HMMT 二月 2023 · 冲刺赛 · 第 10 题

HMMT February 2023 — Guts Round — Problem 10

[13] The number 32016000000000001 316990099009901 = 101 is the product of two distinct prime numbers. Compute the smaller of these two primes.

解析

[13] The number 32016000000000001 316990099009901 = 101 is the product of two distinct prime numbers. Compute the smaller of these two primes. Proposed by: Rishabh Das Answer: 4002001 Solution: Let x = 2000, so the numerator is 5 4 2 3 x + x + 1 = ( x + x + 1)( x − x + 1) . 2 5 4 (This latter factorization can be noted by the fact that plugging in ω or ω into x + x + 1 gives 0.) 2 Then x + x + 1 = 4002001 divides the numerator. However, it can easily by checked that 101 doesn’t divide 4002001 (since, for example, 101 - 1 − 20 + 0 − 4), so 4002001 is one of the primes. Then the other one is 3 3 2000 − 2000 + 1 2000 2 ≈ > 2000 ≈ 4002001 , 101 101 so 4002001 is the smaller of the primes.