Remainder Tests 21 - 25

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