1951 AMC 12 Problem 15

Attempt Problem 15 of the 1951 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1951 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

15.

The largest number by which the expression

n^3-n

is divisible for all possible integral values of

n

is:

2

3

4

5

6

Factor

n^3-n

The factors

n-1,n,n+1

are three consecutive integers

We have

n^3-n=n(n-1)(n+1),

a product of three consecutive integers. One is divisible by

3

and at least one is even, so the product is always divisible by

6.

Taking

n=2

gives exactly

6,

so no larger integer always divides it.

Thus, the correct answer is E.

Problem 15 in other years