The function f(n) is defined as the product of all integers from 1 to n, inclusive, and the function g(n) is defined as the product of all odd integers from 1 to n, inclusive, where n is a positive integer. If p is a prime factor of {f(150)/g(150)} + 1, then which of the following must be true?

Answer Choices

Correct answer marked below

p < 10

10 < p < 25

25 < p < 50

50 < p < 75

p > 75

Correct

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Official Explanation

Official

f(150) is just 150!, and when we divide 150! by all of the odd integers up to 150, we end up just with the product of the even integers from 2 to 150 inclusive.

So we want to know about the smallest prime factor of

(150)(148)(146)...(6)(4)(2) + 1

If we factor out one 2 from each term in that big product, we can rewrite it more succinctly:

(150)(148)(146)...(6)(4)(2) + 1 = (2)(75)(2)(74)(2)(73)...(2)(3)(2)(2)(2)(1) + 1 = 2^{75} \times 75! + 1

Now, 75! is divisible by every prime less than 75. So if you take any such prime, say 47, then

2^{75} \times 75!

will be divisible by 47, and

2^{75} \times 75! + 1

will be exactly 1 greater than a multiple of 47.…

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What this question tests

What it tests

Your ability to evaluate and compose functions, including unusual defined operations.

Common trap

Applying function operations in the wrong order or ignoring domain restrictions.