2017 USAJMO Problems/Problem 4
Problem
Are there any triples of positive integers such that is prime that properly divides the positive number ?
Solution
Yes. Let . Also define . We want to divide the positive number . This equality can be verified by expanding the righthand side. Because will be trivially positive if is non-negative, we can just assume that . Analyzing the structure of , we see that ,, and must be or mod , or will not be prime (divisibility by and ). Thus, we can guess any ,, and which satisfies those constraints. For example, ,, and works. is prime, and it divides the positive number .
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See also
2017 USAJMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |