SAT Math level 1 subject test - help with a question

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1

" What is the least positive integer that has the same number of positive factors as 175?

(A) 8
(B) 10
© 12
(D) 16
(E) 18

Explanation: The number 175 is equal to 5 times 5 times 7. It has a total of 6 positive factors: 1, 5, 7, 25, 35, and 175. The numbers with 6 positive factors are of the form (P1)^2 times P2 or P^5. The least number of the first form is (2^2) times 3 = 12. The least number of the second form is 2^5 = 32. Thus, the answer is 12, which has the 6 positive factors 1, 2, 3, 4, 6, and 12.

You could also test each of the choices, but that may take additional time."

I don’t really understand the method they use to explain how to solve this question. Help please?

2

In general, if N = p1^a1 * p2^a2 * … * pk^ak where p1, …, pk are distinct primes, then the number of divisors of N is (a1+1)(a2+1)…(ak+1) (basically, add 1 to each of the exponents in the prime factorization of N and multiply the exponent).

This fact simply comes from the fact that any divisor of N is of the form

p1^d1 * p2^d2 * … * pk^dk

where

0 ≤ d1≤ a1
0 ≤ d2 ≤ a2
…
0 ≤ dk ≤ ak

So we have a1 + 1 choices for d1, a2 + 1 choices for d2, etc. Multiply them to get the # of factors of N.