2003 AMC 12B Problem 12

Question

What is the largest integer that is a divisor of

\begin{aligned} &(n + 1)(n + 3)(n + 5) \\ &\quad {}\cdot (n + 7)(n + 9) \end{aligned}

for all positive even integers

n?

3

5

11

15

165

For even

n,

the factors are five consecutive odd numbers

Among any five consecutive odd numbers, one is a multiple of

3

and one is a multiple of

5

For even

n,

the five factors are consecutive odd numbers. Among any five consecutive odd numbers, at least one is divisible by

3

and exactly one by

5,

so the product is always divisible by

15.

No larger divisor always works: the products for

n = 10

and

n = 20

are

11 \cdot 13 \cdot 15 \cdot 17 \cdot 19

and

21 \cdot 23 \cdot 25 \cdot 27 \cdot 29,

whose greatest common divisor is

15.

Thus, the correct answer is D.

Problem 12 in other years