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Two integers m and n are selected independently and uniformly from the set {1, 2,..., 100}. What is the probability that the number 7^m + 7ⁿ is divisible by 5?
- 1/4
- 1/7
- 1/8
- 1/49
Correct answer: 1/4
Solution
7^k mod 5 cycles through 2,4,3,1. The sum 7^m + 7^n is divisible by 5 only for residue pairs (2,3),(3,2),(4,1),(1,4). For each m, exactly 1 of 4 residue classes of n works, giving probability 1/4.
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