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Square Product

Problem

Given that [n(n+1)(n+2)]2 = 3039162537*6, find the value of *.


Solution

In any three consecutive integers, n, n+1, n+2, at least one of the numbers will be even, and one of them will be a multiple of 3. Hence the product, n(n+1)(n+2), will be even and divisible by 3. Furthermore, the square of an even number will be divisible by 4, and the square of a multiple of 3 will be divisible by 9.

If a number is divisible by 9, the sum of the digits will also be divisible by 9: 3 + 3 + 9 + 1 + 6 + 2 + 5 + 3 + 7 + 6 = 45, so the value of * must be 0 or 9.

However, if the number is divisible by 4, the last two digits (either 06 or 96) must be divisible by 4. Hence the value of * is 9.

Find the value of n.
How would you solve the equation, [n(n+1)(n+2)]2 = k, in general?

Problem ID: 178 (Oct 2004)     Difficulty: 2 Star

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