Mathematics
Problem of The Day
2024
111

111

For every natural number n, prove that the expression n5 - 5n3 + 4n is always divisible by 120.

Answer
SOLUTIOn

We begin by factoring the expression n5 - 5n3 + 4n as n(n2−1)(n2−4)=(n−2)(n−1)n(n+1)(n+2).

Notice that among five consecutive integers, one is divisible by 5, at least one is divisible by 3, and two are consecutive even numbers, one of which is divisible by 2 and the other by 4.

Therefore, the expression is divisible by 2×4×3×5=1202×4×3×5=120.

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