HMMT 二月 2018 · ALGNT 赛 · 第 4 题

HMMT February 2018 — ALGNT Round — Problem 4

Distinct prime numbers p, q, r satisfy the equation 2 pqr + 50 pq = 7 pqr + 55 pr = 8 pqr + 12 qr = A for some positive integer A . What is A ? 101 x − 1

解析

Distinct prime numbers p, q, r satisfy the equation 2 pqr + 50 pq = 7 pqr + 55 pr = 8 pqr + 12 qr = A for some positive integer A . What is A ? Proposed by: Kevin Sun Answer: 1980 A Note that A is a multiple of p , q , and r , so K = is an integer. Dividing through, we have that pqr 12 55 50 K = 8 + = 7 + = 2 + . p q r Then p ∈ { 2 , 3 } , q ∈ { 5 , 11 } , and r ∈ { 2 , 5 } . These values give K ∈ { 14 , 12 } , K ∈ { 18 , 12 } , and K ∈ { 27 , 12 } , giving K = 12 and ( p, q, r ) = (3 , 11 , 5). We can then compute A = pqr · K = 3 · 11 · 5 · 12 = 1980. 101 x − 1