For each integer N>1, define positive integers to be congruent if they leave the same nonnegative remainder when divided by N. If 69, 90, and 125 are congruent in one such system, then in that same system, 81 is congruent to:

1971 AMC 12 Problem 12

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12.

For each integer

N\gt1,

define positive integers to be congruent if they leave the same nonnegative remainder when divided by

N.

69,

90,

and

125

are congruent in one such system, then in that same system,

81

is congruent to:

3

4

5

7

8

The modulus divides the difference of any two congruent integers

Use the differences

21

and

35

The modulus

N

divides both

90-69=21

and

125-90=35.

Since

N\gt1,

this forces

N=7.

Then

81\equiv4\pmod7.

Therefore, the correct answer is B.