HMMT 二月 2002 · 冲刺赛 · 第 54 题

HMMT February 2002 — Guts Round — Problem 54

[10] How many pairs of integers ( a, b ), with 1 ≤ a ≤ b ≤ 60, have the property that b is divisible by a and b + 1 is divisible by a + 1?

解析

How many pairs of integers ( a, b ), with 1 ≤ a ≤ b ≤ 60, have the property that b is divisible by a and b + 1 is divisible by a + 1? Solution: The divisibility condition is equivalent to b − a being divisible by both a and a + 1, or, equivalently (since these are relatively prime), by a ( a + 1). Any b satisfying the condition is automatically ≥ a , so it suffices to count the number of values b − a ∈ { 1 − a, 2 − a, . . . , 60 − a } that are divisible by a ( a + 1) and sum over all a . The number of such values will be precisely 60 / [ a ( a + 1)] whenever this quantity is an integer, which fortunately happens for every a ≤ 5; we count: a = 1 gives 30 values of b ; 13 a = 2 gives 10 values of b ; a = 3 gives 5 values of b ; a = 4 gives 3 values of b ; a = 5 gives 2 values of b ; a = 6 gives 2 values ( b = 6 or 48); any a ≥ 7 gives only one value, namely b = a , since b > a implies b ≥ a + a ( a + 1) > 60. Adding these up, we get a total of 106 pairs.