1971 AHSME Problems/Problem 12
Problem
For each integer , there is a mathematical system in which two or more positive integers are defined to be congruent if they leave the same non-negative remainder when divided by If , and are congruent in one such system, then in that same system, is congruent to
Solution
The "mathematical system" being alluded to is modulo , so we shall be using such notation for convenience. From the problem, we know that , so the difference between each of these numbers must be a multiple of . , and . The only common factors between these two numbers are and , but we know from the problem that . Thus, . Now, , so our answer is .
See Also
1971 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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