1993 AIME Problem 6

Question

What is the smallest positive integer that can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?

Answer

Answer: 495

Concepts:arithmetic sequence least common multiple modular arithmetic

Difficulty rating: 1740

Small hint:

A sum of an odd number of consecutive integers is divisible by the number of terms

Big hint:

A sum of ten consecutive integers is congruent to

5\pmod {10}

Solution:

The sums of

9

and

11

consecutive integers are divisible by

9

and

11,

so the desired number is a multiple of

99.

A sum of

10

consecutive integers has the form

10a+45,

hence is congruent to

5\pmod {10}.

The first multiple of

99

ending in

5

is

5\cdot99=495,

and each of the three required representations then exists.