1968 AMC 12 Problem 15

Attempt Problem 15 of the 1968 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 1968 AMC 12 solutions, or check the answer key.

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

15.

Let

P

be the product of any three consecutive positive odd integers. The largest integer dividing all such

P

is:

15

6

5

3

1

Among three consecutive odd integers, inspect residues modulo

3

Use two examples to rule out every larger common divisor

Three consecutive odd integers occupy three consecutive residues modulo

3,

so one is divisible by

3.

Thus every product is divisible by

3.

The products

1\cdot3\cdot5=15

and

7\cdot9\cdot11=693

have greatest common divisor

3,

so no larger integer always divides the product.

Therefore, the correct answer is D.

Problem 15 in other years