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