PUMaC 2013 · 数论（A 组） · 第 3 题

PUMaC 2013 — Number Theory (Division A) — Problem 3

[ 4 ] Let A be the greatest possible value of a product of positive integers that sums to 2014. Compute the sum of all bases and exponents in the prime factorization of A . For example, if 5 A = 7 · 11 , the answer would be 7 + 11 + 5 = 23. 10 45 30 30

解析

[ 4 ] Let A be the greatest possible value of a product of positive integers that sums to 2014. Compute the sum of all bases and exponents in the prime factorization of A . For example, if 5 A = 7 · 11 , the answer would be 7 + 11 + 5 = 23. 1 n − 1 n +1 Solution Note that if we have a large enough n , odd n ≥ 5, break it as n = + . This 2 2 n n product is larger than n For even n ≥ 4, break it as n = + . This product is larger than n . 2 2 3 2 Furthermore, noting that 2 < 3 , we should have at most two 2’s. So the optimum case is (2)(2) + (670)(3) = 2014. The answer is 677. 10 45 30 30