1960 AMC 12 Problem 25

Attempt Problem 25 of the 1960 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 1960 AMC 12 solutions, or check the answer key.

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

25.

Let

m

and

n

be any two odd numbers, with

n

less than

m.

The largest integer which divides all possible numbers of the form

m^2-n^2

is:

2

4

6

8

16

Factor

m^2-n^2

into two even factors

Among

m-n

and

m+n,

one is divisible by

4

; then test a small pair

We have

m^2-n^2=(m-n)(m+n).

Both factors are even, and one is divisible by

4,

so every such difference is divisible by

8.

Taking

m=3,n=1

gives

m^2-n^2=8,

proving that no larger integer always divides it.

Thus, the correct answer is D.

Problem 25 in other years