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