Which one of the following numbers is exactly divisible by (11^13 + 1)?
Options:
(A) 11^26 + 1
(B) 11^33 + 1
(C) 11^39 − 1
(D) 11^52 − 1
Official Answer Key:
D
Step-by-Step Solution & Explanation
Let x = 11. Since x ≡ −1 (mod x^13 + 1), any even power of x^13 is ≡ 1 (mod x^13+1). x^52 = (x^13)^4 ≡ (−1)^4 = 1 (mod x^13+1), so x^52 − 1 ≡ 0 — divisible. Checking distractors: x^26 = (x^13)^2 ≡ 1, so x^26+1 ≡ 2 (not divisible, rules out A). x^33 is not an integer multiple of 13 so no clean reduction applies (rules out B). x^39=(x^13)^3≡(−1)^3=−1, so x^39−1≡−2 (not divisible, rules out C). Hence (D) is correct.
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