2005 AMC 10A Problem 24

Question

For each positive integer

m \gt 1,

let

P(m)

denote the greatest prime factor of

m.

For how many positive integers

n

is it true that both

P(n) = \sqrt{n}

and

P(n + 48) = \sqrt{n + 48}?

0

1

3

4

5

P(n) = \sqrt{n}

forces

n

to be the square of a prime

Write

n = q^2

and

n + 48 = p^2;

then

48 = p^2 - q^2 = (p - q)(p + q)

The condition

P(n) = \sqrt{n}

means

n

is the square of a prime

q,

and likewise

n + 48 = p^2

for a prime

p.

Then

48 = p^2 - q^2 = (p - q)(p + q).

Checking the same-parity factorizations of

48,

only

p - q = 2,\ p + q = 24

yields primes, giving

(p, q) = (13, 11)

and

n = 121.

So there is exactly one such

n.

Thus, the correct answer is B.

Problem 24 in other years