21 - (1, 10, -5, -8, 4, -2) Period: 6 digits
Recurring numbers: 1, 10, -5, -8, 4, -2
This is the minimum magnitude sequence.
(1, 10, 16, 13, 4, 19) Period : 6 digits
Recurring numbers: 1, 10, 16, 13, 4, 19
This is the positive sequence.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 21.
Example: What is the remainder when 2306451879 is divided by 21?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x 10 = 70
Multiplication of the third rightmost digit = 8 x -5 = -40
Multiplication of the fourth rightmost digit = 1 x -8 = -8
Fifth rightmost digit = 5 x 4 = 20
Sixth rightmost digit = 4 x -2 = -8
Seventh rightmost digit = 6 x 1 = 6
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x -5 = -15
Tenth rightmost digit = 2 x -8 = -16
Sum = 18
18 modulus 21 = 18
Remainder = 18
22 - (1, 10, -10, 10, -10, and the cycle goes on) Period: 2 digits
Recurring numbers: 10, -10
This is the minimum magnitude sequence.
(1, 10, 12, 10, 12, and the cycle goes on) Period: 2 digits
Recurring numbers: 10, 12
This is the positive sequence.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 22.
Example: What is the remainder when 2306451879 is divided by 22?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x 10 = 70
Multiplication of the third rightmost digit = 8 x -10 = -80
Multiplication of the fourth rightmost digit = 1 x 10 = 10
Fifth rightmost digit = 5 x -10 = -50
Sixth rightmost digit = 4 x 10 = 40
Seventh rightmost digit = 6 x -10 = -60
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x -10 = -30
Tenth rightmost digit = 2 x 10 = 20
Sum = -71
-71 modulus 22 = 17
Remainder = 17
23 - (1, 10, 8, 11, -5, -4, 6, -9, 2, -3, -7, -1, -10, -8, -11, 5, 4, -6, 9, -2, 3, 7, and the cycle goes on) Period: 22 digits
Recurring numbers: 1, 10, 8, 11, -5, -4, 6, -9, 2, -3, -7, -1, -10, -8, -11, 5, 4, -6, 9, -2, 3, 7
This is the minimum magnitude sequence.
(1, 10, 8, 11, 18, 19, 6, 14, 2, 20, 16, 22, 13, 15, 12, 5, 4, 17, 9, 21, 3, 7, and the cycle goes on) Period: 22 digits
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 23.
Example: What is the remainder when 2306451879 is divided by 23?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x 10 = 70
Multiplication of the third rightmost digit = 8 x 8 = 64
Multiplication of the fourth rightmost digit = 1 x 11 = 11
Fifth rightmost digit = 5 x -5 = -25
Sixth rightmost digit = 4 x -4 = -16
Seventh rightmost digit = 6 x 6 = 36
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x 2 = 6
Tenth rightmost digit = 2 x -3 = -6
Sum = 149
149 modulus 23 = 11
Remainder = 11
24 - (1, 10, 4, -8, -8, -8, -8, and so on) No period.
Recurring number: -8 starting from fourth digit of the number.
This is the minimum magnitude sequence.
(1, 10, 4, 16, 16, 16, 16, and so on) No period.
Recurring number: 16 starting from fourth digit of the number.
This is the positive sequence.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 24.
Example: What is the remainder when 2306451879 is divided by 24?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x 10 = 70
Multiplication of the third rightmost digit = 8 x 4 = 32
Multiplication of the fourth rightmost digit = 1 x -8 = -8
Fifth rightmost digit = 5 x -8 = -40
Sixth rightmost digit = 4 x -8 = -32
Seventh rightmost digit = 6 x -8 = -48
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x -8 = -24
Tenth rightmost digit = 2 x -8 = -16
Sum = -57
-57 modulus 24 = 15
Remainder = 15
25 - No period, No recurring seqeunce.
Take the last two digits of a very big number. After that, take that number modulus 25.
Example: What is the remainder when 2306451879 is divided by 24?
Last two digits: 79
79 modulus 25 = 4
Remainder = 4
Saturday, May 16, 2009
Tuesday, May 12, 2009
Remainder Tests 16 - 20
16 - (1, -6, 4, 8) No period. No recurring numbers.
The sequence only applies to the four extreme right digits. In addition, this sequence is the minimum magnitude sequence.
(1, 10, 4, 8) No period. No recurring numbers.
The sequence only applies to the four extreme right digits. In addition, this sequence is the positive sequence.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 16.
Example: What is the remainder when 2306451879 is divided by 16?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -6 = -42
Multiplication of the third rightmost digit = 8 x 4 = 32
Multiplication of the fourth rightmost digit = 1 x 8 = 8
Sum = 7
7 modulus 16 = 7
Remainder = 7
17 - (1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, -4, -6, 8, -5) Period: 16 digits. Recurring numbers: 1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, -4, -6, 8, -5
This sequence is the minimum magnitude sequence.
(1, 10, 15, 14, 4, 6, 9, 5, 16, 7, 2, 3, 13, 11, 8, 12) Period: 16 digits. Recurring numbers: 1, 10, 15, 14, 4, 6, 9, 5, 16, 7, 2, 3, 13, 11, 8, 12
This sequence is the positive sequence.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 17.
Example: What is the remainder when 2306451879 is divided by 17?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -7 = -49
Multiplication of the third rightmost digit = 8 x -2 = -16
Multiplication of the fourth rightmost digit = 1 x -3 = -3
Fifth rightmost digit = 5 x 4 = 20
Sixth rightmost digit = 4 x 6 = 24
Seventh rightmost digit = 6 x -8 = -48
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x -1 = -3
Tenth rightmost digit = 2 x 7 = 14
Sum = -52
-52 modulus 17 = 16
Remainder = 16
18 - (1, -8, -8, -8 and -8 goes on for the rest of the digits) No period.
Recurring number: -8
This is the minimum magnitude sequence.
(1, 10, 10, 10, 10 and 10 goes on for the rest of the digits) No period.
Recurring number: 10
This is the positive sequence.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 18.
Example: What is the remainder when 2306451879 is divided by 18?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -8 = -56
Multiplication of the third rightmost digit = 8 x -8 = -64
Multiplication of the fourth rightmost digit = 1 x -8 = -8
Fifth rightmost digit = 5 x -8 = -40
Sixth rightmost digit = 4 x -8 = -32
Seventh rightmost digit = 6 x -8 = -48
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x -8 = -24
Tenth rightmost digit = 2 x -8 = -16
Sum = -279
-279 modulus 18 = 9
Remainder = 9
19 - (1, -9, 5, -7, 6, 3, -8, -4, -2, -1, 9, -5, 7, -6, -3, 8, 4, 2)
Period: 18 digits.
Recurring numbers: 1, -9, 5, -7, 6, 3, -8, -4, -2, -1, 9, -5, 7, -6, -3, 8, 4, 2
This is the minimum magnitude sequence.
(1, 10, 5, 12, 6, 3, 11, 15, 17, 18, 9, 14, 7, 13, 16, 8, 4, 2)
Period: 18 digits.
Recurring numbers: 1, -9, 5, -7, 6, 3, -8, -4, -2, -1, 9, -5, 7, -6, -3, 8, 4, 2
This is the positive sequence.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 19.
Example: What is the remainder when 2306451879 is divided by 19?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -9 = -63
Multiplication of the third rightmost digit = 8 x 5 = 40
Multiplication of the fourth rightmost digit = 1 x -7 = -7
Fifth rightmost digit = 5 x 6 = 30
Sixth rightmost digit = 4 x 3 = 12
Seventh rightmost digit = 6 x -8 = -48
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x -2 = -6
Tenth rightmost digit = 2 x -1 = -2
Sum = -35
-35 modulus 19 = 3
Remainder = 3
20 - No period, no recurring numbers, no sequence.
Take the last two digits of a very big number. After that, take that number modulus 20.
Example: What is the remainder when 2306451879 is divided by 20?
Last two digits: 79
79 modulus 20 = 19
Remainder = 19
The sequence only applies to the four extreme right digits. In addition, this sequence is the minimum magnitude sequence.
(1, 10, 4, 8) No period. No recurring numbers.
The sequence only applies to the four extreme right digits. In addition, this sequence is the positive sequence.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 16.
Example: What is the remainder when 2306451879 is divided by 16?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -6 = -42
Multiplication of the third rightmost digit = 8 x 4 = 32
Multiplication of the fourth rightmost digit = 1 x 8 = 8
Sum = 7
7 modulus 16 = 7
Remainder = 7
17 - (1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, -4, -6, 8, -5) Period: 16 digits. Recurring numbers: 1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, -4, -6, 8, -5
This sequence is the minimum magnitude sequence.
(1, 10, 15, 14, 4, 6, 9, 5, 16, 7, 2, 3, 13, 11, 8, 12) Period: 16 digits. Recurring numbers: 1, 10, 15, 14, 4, 6, 9, 5, 16, 7, 2, 3, 13, 11, 8, 12
This sequence is the positive sequence.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 17.
Example: What is the remainder when 2306451879 is divided by 17?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -7 = -49
Multiplication of the third rightmost digit = 8 x -2 = -16
Multiplication of the fourth rightmost digit = 1 x -3 = -3
Fifth rightmost digit = 5 x 4 = 20
Sixth rightmost digit = 4 x 6 = 24
Seventh rightmost digit = 6 x -8 = -48
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x -1 = -3
Tenth rightmost digit = 2 x 7 = 14
Sum = -52
-52 modulus 17 = 16
Remainder = 16
18 - (1, -8, -8, -8 and -8 goes on for the rest of the digits) No period.
Recurring number: -8
This is the minimum magnitude sequence.
(1, 10, 10, 10, 10 and 10 goes on for the rest of the digits) No period.
Recurring number: 10
This is the positive sequence.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 18.
Example: What is the remainder when 2306451879 is divided by 18?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -8 = -56
Multiplication of the third rightmost digit = 8 x -8 = -64
Multiplication of the fourth rightmost digit = 1 x -8 = -8
Fifth rightmost digit = 5 x -8 = -40
Sixth rightmost digit = 4 x -8 = -32
Seventh rightmost digit = 6 x -8 = -48
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x -8 = -24
Tenth rightmost digit = 2 x -8 = -16
Sum = -279
-279 modulus 18 = 9
Remainder = 9
19 - (1, -9, 5, -7, 6, 3, -8, -4, -2, -1, 9, -5, 7, -6, -3, 8, 4, 2)
Period: 18 digits.
Recurring numbers: 1, -9, 5, -7, 6, 3, -8, -4, -2, -1, 9, -5, 7, -6, -3, 8, 4, 2
This is the minimum magnitude sequence.
(1, 10, 5, 12, 6, 3, 11, 15, 17, 18, 9, 14, 7, 13, 16, 8, 4, 2)
Period: 18 digits.
Recurring numbers: 1, -9, 5, -7, 6, 3, -8, -4, -2, -1, 9, -5, 7, -6, -3, 8, 4, 2
This is the positive sequence.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 19.
Example: What is the remainder when 2306451879 is divided by 19?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -9 = -63
Multiplication of the third rightmost digit = 8 x 5 = 40
Multiplication of the fourth rightmost digit = 1 x -7 = -7
Fifth rightmost digit = 5 x 6 = 30
Sixth rightmost digit = 4 x 3 = 12
Seventh rightmost digit = 6 x -8 = -48
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x -2 = -6
Tenth rightmost digit = 2 x -1 = -2
Sum = -35
-35 modulus 19 = 3
Remainder = 3
20 - No period, no recurring numbers, no sequence.
Take the last two digits of a very big number. After that, take that number modulus 20.
Example: What is the remainder when 2306451879 is divided by 20?
Last two digits: 79
79 modulus 20 = 19
Remainder = 19
Thursday, May 7, 2009
Remainder Tests 11 - 15
Starting from 11, here it goes.
11 - (1, -1, and the cycle goes on for the next two digits) Period: 2 digits. Recurring numbers: 1, -1
This is the minimum magnitude sequence
(1, 10, and the cycle goes on for the next two) Period: 2 digits. Recurring numbers: 1, 10
This is the positive sequence.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 11.
Example: What is the remainder when 2306451879 is divided by 11?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -1 = -7
Multiplication of the third rightmost digit = 8 x 1 = 8
Multiplication of the fourth rightmost digit = 1 x -1 = -1
Fifth rightmost digit = 5 x 1 = 5
Sixth rightmost digit = 4 x -1 = -4
Seventh rightmost digit = 6 x 1 = 6
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x 1 = 3
Tenth rightmost digit = 2 x -1 = -2
Sum = 17
17 modulus 11 = 6
Remainder = 6
12 - (1, -2, 4, 4, and 4 goes on) No period.
Minimum magnitude sequence
(1, 10, 4, 4, and 4 goes on) No period.
Positive sequence
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 12.
Example: What is the remainder when 2306451879 is divided by 12?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -2 = -14
Multiplication of the third rightmost digit = 8 x 4 = 32
Multiplication of the fourth rightmost digit = 1 x 4 = 4
Fifth rightmost digit = 5 x 4 = 20
Sixth rightmost digit = 4 x 4 = 16
Seventh rightmost digit = 6 x 4 = 24
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x 4 = 12
Tenth rightmost digit = 2 x 4 = 8
Sum = 111
111 modulus 12 = 3
Remainder = 3
13 - (1, -3, -4, -1, 3, 4, and the cycle goes on for the next 6 digits) Period: 6 digits.
Minimum magnitude sequence
(1, 10, 9, 12, 3, 4, and the cycle goes on for the next 6 digits) Period: 6 digits
Positive sequence
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 13.
Example: What is the remainder when 2306451879 is divided by 13?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -3 = -21
Multiplication of the third rightmost digit = 8 x -4 = -32
Multiplication of the fourth rightmost digit = 1 x -1 = -1
Fifth rightmost digit = 5 x 3 = 15
Sixth rightmost digit = 4 x 4 = 16
Seventh rightmost digit = 6 x 1 = 6
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x -4 = -12
Tenth rightmost digit = 2 x -1 = -2
Sum = -22
-22 modulus 13 = 4
Remainder = 4
14 - (1, -4, 2, 6, 4, -2, -6, and the cycle continues for the six digits) Period: 6 digits. Recurring digits: -4, 2, 6, 4, -2, -6
This is the minimum magnitude sequence.
(1, 10, 2, 6, 4, 12, 8, and the cycle continues for the next six digits) Period: 6 digits. Recurring digits: 10, 2, 6, 4, 12, 8
This is the positive sequence.
1 for both sequences only applies only to the first extreme right digit.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 14.
Example: What is the remainder when 2306451879 is divided by 14?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -4 = -28
Multiplication of the third rightmost digit = 8 x 2 = 16
Multiplication of the fourth rightmost digit = 1 x 6 = 6
Fifth rightmost digit = 5 x 4 = 20
Sixth rightmost digit = 4 x -2 = -8
Seventh rightmost digit = 6 x -6 = -36
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x 2 = 6
Tenth rightmost digit = 2 x 6 = 12
Sum = -3
-3 modulus 14 = 11
Remainder = 11
15 - (1, -5, -5, -5, and -5 continues for the rest of the digits) No period. Recurring number: -5
This is the minimum magnitude sequence.
(1, 10, 10, 10, and 10 continues for the rest of the digits) No period. Recurring number: 10
This is the positive sequence.
1 for both sequences only applies only to the first extreme right digit.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 15.
Example: What is the remainder when 2306451879 is divided by 15?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -5 = -35
Multiplication of the third rightmost digit = 8 x -5 = -40
Multiplication of the fourth rightmost digit = 1 x -5 = -5
Fifth rightmost digit = 5 x -5 = -25
Sixth rightmost digit = 4 x -5 = -20
Seventh rightmost digit = 6 x -5 = -30
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x -5 = -15
Tenth rightmost digit = 2 x -5 = -10
Sum = -171
-171 modulus 15 = 9
Remainder = 9
11 - (1, -1, and the cycle goes on for the next two digits) Period: 2 digits. Recurring numbers: 1, -1
This is the minimum magnitude sequence
(1, 10, and the cycle goes on for the next two) Period: 2 digits. Recurring numbers: 1, 10
This is the positive sequence.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 11.
Example: What is the remainder when 2306451879 is divided by 11?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -1 = -7
Multiplication of the third rightmost digit = 8 x 1 = 8
Multiplication of the fourth rightmost digit = 1 x -1 = -1
Fifth rightmost digit = 5 x 1 = 5
Sixth rightmost digit = 4 x -1 = -4
Seventh rightmost digit = 6 x 1 = 6
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x 1 = 3
Tenth rightmost digit = 2 x -1 = -2
Sum = 17
17 modulus 11 = 6
Remainder = 6
12 - (1, -2, 4, 4, and 4 goes on) No period.
Minimum magnitude sequence
(1, 10, 4, 4, and 4 goes on) No period.
Positive sequence
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 12.
Example: What is the remainder when 2306451879 is divided by 12?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -2 = -14
Multiplication of the third rightmost digit = 8 x 4 = 32
Multiplication of the fourth rightmost digit = 1 x 4 = 4
Fifth rightmost digit = 5 x 4 = 20
Sixth rightmost digit = 4 x 4 = 16
Seventh rightmost digit = 6 x 4 = 24
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x 4 = 12
Tenth rightmost digit = 2 x 4 = 8
Sum = 111
111 modulus 12 = 3
Remainder = 3
13 - (1, -3, -4, -1, 3, 4, and the cycle goes on for the next 6 digits) Period: 6 digits.
Minimum magnitude sequence
(1, 10, 9, 12, 3, 4, and the cycle goes on for the next 6 digits) Period: 6 digits
Positive sequence
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 13.
Example: What is the remainder when 2306451879 is divided by 13?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -3 = -21
Multiplication of the third rightmost digit = 8 x -4 = -32
Multiplication of the fourth rightmost digit = 1 x -1 = -1
Fifth rightmost digit = 5 x 3 = 15
Sixth rightmost digit = 4 x 4 = 16
Seventh rightmost digit = 6 x 1 = 6
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x -4 = -12
Tenth rightmost digit = 2 x -1 = -2
Sum = -22
-22 modulus 13 = 4
Remainder = 4
14 - (1, -4, 2, 6, 4, -2, -6, and the cycle continues for the six digits) Period: 6 digits. Recurring digits: -4, 2, 6, 4, -2, -6
This is the minimum magnitude sequence.
(1, 10, 2, 6, 4, 12, 8, and the cycle continues for the next six digits) Period: 6 digits. Recurring digits: 10, 2, 6, 4, 12, 8
This is the positive sequence.
1 for both sequences only applies only to the first extreme right digit.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 14.
Example: What is the remainder when 2306451879 is divided by 14?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -4 = -28
Multiplication of the third rightmost digit = 8 x 2 = 16
Multiplication of the fourth rightmost digit = 1 x 6 = 6
Fifth rightmost digit = 5 x 4 = 20
Sixth rightmost digit = 4 x -2 = -8
Seventh rightmost digit = 6 x -6 = -36
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x 2 = 6
Tenth rightmost digit = 2 x 6 = 12
Sum = -3
-3 modulus 14 = 11
Remainder = 11
15 - (1, -5, -5, -5, and -5 continues for the rest of the digits) No period. Recurring number: -5
This is the minimum magnitude sequence.
(1, 10, 10, 10, and 10 continues for the rest of the digits) No period. Recurring number: 10
This is the positive sequence.
1 for both sequences only applies only to the first extreme right digit.
Multiply the rightmost digit by the leftmost digit in the sequence. Next, multiply the second rightmost digit by the second leftmost digit in the sequence. It goes on for the subsequent digits. After that, take the sum of the all the computed values. Finally, take the sum modulus of 15.
Example: What is the remainder when 2306451879 is divided by 15?
Multiplication of the rightmost digit = 9 x 1 = 9
Multiplication of the second rightmost digit = 7 x -5 = -35
Multiplication of the third rightmost digit = 8 x -5 = -40
Multiplication of the fourth rightmost digit = 1 x -5 = -5
Fifth rightmost digit = 5 x -5 = -25
Sixth rightmost digit = 4 x -5 = -20
Seventh rightmost digit = 6 x -5 = -30
Eighth rightmost digit = 0
Ninth rightmost digit = 3 x -5 = -15
Tenth rightmost digit = 2 x -5 = -10
Sum = -171
-171 modulus 15 = 9
Remainder = 9
Tuesday, May 5, 2009
Remainder Tests 2 - 10
Remainder Tests from 2 - 10
2 - Take the right most digit of a very big number and take its modulus by 2. You can ignore the rest of the digts. The remainder will be the same as the remainder when the whole number divides by 2.
Example: What is the remainder when 1036125837 is divided by 2?
The last digit of the number is 7.
7 modulus 2 = 1.
Remainder = 1
3 - Take the sum of all the digits of a very big number. Next, after finding the sum, take its modulus by 3. The remainder will be the same as finding the remainder when the whole number divides by 3.
Example: What is the remainder when 1036125837 is divided by 3?
Sum = 1 + 0 + 3 + 6 + 1 + 2 + 5 + 8 + 3 + 7 = 36
36 modulus 3 = 0.
Remainder = 0
4 - (1, 2) No period.
The sequence that is shown on the left is only applied on the two right most digits only. Multiply the right most digit by 1 and multiply the second right most digit by 2. Add both of them together and take the modulus of 4.
Example: What is the remainder when 1036125837 is divided by 4?
The last two digits of the number is 37.
Multiplication of the right most digit: 7 x 1 = 7
Multiplication of the left most digit: 3 x 2 = 6
Sum = 7 + 6 = 13
13 modulus 4 = 1, Remainder = 1
5 - Take the right most digit and take its modulus of 5.
Example: What is the remainder when 1036125837 is divided by 5?
Rightmost digit = 7
7 modulus 5 = 2, Remainder = 2
6 - (1, -2, -2, -2, -2, and -2 goes on for the rest)No period
Minimum magnitude sequence
(1, 4, 4, 4, 4, and 4 goes on for the rest)
Positive sequence
Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on. Next, compute the sum of all the values and take the modulus of 6.
Example: What is the remainder when 1036125837 is divided by 6?
Multiplication of the rightmost digit = 1 x 7 = 7
Multiplication of the second rightmost digit = 3 x -2 = -6
Third rightmost digit = -16, Fourth rightmost digit = -10,
Fifth rightmost digit = -4, Sixth rightmost digit = -2
Seventh rightmost digit = -12, Eighth rightmost digit = -6
Ninth rightmost digit = 0, Tenth rightmost digit = -2
Sum = -51
-51 modulus 6 = 3
Remainder = 3
7 - (1, 3, 2, -1, -3, -2, cycle repeats for the next six digits) Period: 6 digits.
Recurring numbers: 1, 3, 2, -1, -3, -2
Minimum magnitude sequence
(1, 3, 2, 6, 4, 5, cycle repeats for the next six digits) Period: 6 digits.
Recurring numbers: 1, 3, 2, 6, 4, 5
Positive sequence
Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on and so for. Next, compute the sum of all the values and take the modulus of 7.
Example: What is the remainder when 1036125837 is divided by 7?
Multiplication of the rightmost digit = 1 x 7 = 7
Multiplication of the second rightmost digit = 3 x 3 = 9
Third rightmost digit = 8 x 2 = 16, Fourth rightmost digit = 5 x -1 = -5,
Fifth rightmost digit = 2 x -3 = -6, Sixth rightmost digit = 1 x -2 = -2
Seventh rightmost digit = 6 x 1 = 6, Eighth rightmost digit = 3 x 3 = 9
Ninth rightmost digit = 0, Tenth rightmost digit = 1 x -1 = -1
Sum = 33
33 modulus 7 = 5
Remainder = 5
8 - (1, 2, 4) No period.
The sequence shown on the left is only appliable on the three extreme right digits of a very big number. The rest of the digits can be ignored.
Multiply the right most digit by 1 and multiply the second right most digit by 2 and multiply the third rightmost digit by 4. Add all of them together and take the modulus of 8.
Example: What is the remainder when 1036125837 is divided by 8?
Multiplication of the rightmost digit = 7 x 1 = 7
Multiplication of the second rightmost digit = 3 x 2 = 6
Multiplication of the third rightmost digit = 8 x 4 = 32
Sum = 45
45 modulus 8 = 5
Remainder = 5
9 - It is similar to the remainder test of 3. Take the sum of all the digits of a very big number. Next, after finding the sum, take its modulus by 9. The remainder will be the same as finding the remainder when the whole number divides by 9.
Example: What is the remainder when 1036125837 is divided by 9?
Sum = 1 + 0 + 3 + 6 + 1 + 2 + 5 + 8 + 3 + 7 = 36
36 modulus 9 = 0.
Remainder = 0
10 - Take the rightmost digit of a very large number. The remainder is simply that digit itself. lol.
Example: What is the remainder when 1036125837 is divided by 10?
Rightmost digit = 7
Remainder = 7
This ends this post from 2 - 10.
2 - Take the right most digit of a very big number and take its modulus by 2. You can ignore the rest of the digts. The remainder will be the same as the remainder when the whole number divides by 2.
Example: What is the remainder when 1036125837 is divided by 2?
The last digit of the number is 7.
7 modulus 2 = 1.
Remainder = 1
3 - Take the sum of all the digits of a very big number. Next, after finding the sum, take its modulus by 3. The remainder will be the same as finding the remainder when the whole number divides by 3.
Example: What is the remainder when 1036125837 is divided by 3?
Sum = 1 + 0 + 3 + 6 + 1 + 2 + 5 + 8 + 3 + 7 = 36
36 modulus 3 = 0.
Remainder = 0
4 - (1, 2) No period.
The sequence that is shown on the left is only applied on the two right most digits only. Multiply the right most digit by 1 and multiply the second right most digit by 2. Add both of them together and take the modulus of 4.
Example: What is the remainder when 1036125837 is divided by 4?
The last two digits of the number is 37.
Multiplication of the right most digit: 7 x 1 = 7
Multiplication of the left most digit: 3 x 2 = 6
Sum = 7 + 6 = 13
13 modulus 4 = 1, Remainder = 1
5 - Take the right most digit and take its modulus of 5.
Example: What is the remainder when 1036125837 is divided by 5?
Rightmost digit = 7
7 modulus 5 = 2, Remainder = 2
6 - (1, -2, -2, -2, -2, and -2 goes on for the rest)No period
Minimum magnitude sequence
(1, 4, 4, 4, 4, and 4 goes on for the rest)
Positive sequence
Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on. Next, compute the sum of all the values and take the modulus of 6.
Example: What is the remainder when 1036125837 is divided by 6?
Multiplication of the rightmost digit = 1 x 7 = 7
Multiplication of the second rightmost digit = 3 x -2 = -6
Third rightmost digit = -16, Fourth rightmost digit = -10,
Fifth rightmost digit = -4, Sixth rightmost digit = -2
Seventh rightmost digit = -12, Eighth rightmost digit = -6
Ninth rightmost digit = 0, Tenth rightmost digit = -2
Sum = -51
-51 modulus 6 = 3
Remainder = 3
7 - (1, 3, 2, -1, -3, -2, cycle repeats for the next six digits) Period: 6 digits.
Recurring numbers: 1, 3, 2, -1, -3, -2
Minimum magnitude sequence
(1, 3, 2, 6, 4, 5, cycle repeats for the next six digits) Period: 6 digits.
Recurring numbers: 1, 3, 2, 6, 4, 5
Positive sequence
Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on and so for. Next, compute the sum of all the values and take the modulus of 7.
Example: What is the remainder when 1036125837 is divided by 7?
Multiplication of the rightmost digit = 1 x 7 = 7
Multiplication of the second rightmost digit = 3 x 3 = 9
Third rightmost digit = 8 x 2 = 16, Fourth rightmost digit = 5 x -1 = -5,
Fifth rightmost digit = 2 x -3 = -6, Sixth rightmost digit = 1 x -2 = -2
Seventh rightmost digit = 6 x 1 = 6, Eighth rightmost digit = 3 x 3 = 9
Ninth rightmost digit = 0, Tenth rightmost digit = 1 x -1 = -1
Sum = 33
33 modulus 7 = 5
Remainder = 5
8 - (1, 2, 4) No period.
The sequence shown on the left is only appliable on the three extreme right digits of a very big number. The rest of the digits can be ignored.
Multiply the right most digit by 1 and multiply the second right most digit by 2 and multiply the third rightmost digit by 4. Add all of them together and take the modulus of 8.
Example: What is the remainder when 1036125837 is divided by 8?
Multiplication of the rightmost digit = 7 x 1 = 7
Multiplication of the second rightmost digit = 3 x 2 = 6
Multiplication of the third rightmost digit = 8 x 4 = 32
Sum = 45
45 modulus 8 = 5
Remainder = 5
9 - It is similar to the remainder test of 3. Take the sum of all the digits of a very big number. Next, after finding the sum, take its modulus by 9. The remainder will be the same as finding the remainder when the whole number divides by 9.
Example: What is the remainder when 1036125837 is divided by 9?
Sum = 1 + 0 + 3 + 6 + 1 + 2 + 5 + 8 + 3 + 7 = 36
36 modulus 9 = 0.
Remainder = 0
10 - Take the rightmost digit of a very large number. The remainder is simply that digit itself. lol.
Example: What is the remainder when 1036125837 is divided by 10?
Rightmost digit = 7
Remainder = 7
This ends this post from 2 - 10.
Subscribe to:
Posts (Atom)
