OG 11 - Number Properties

[albertrahul]

How many different positive integers are factors of 441?

(A) 4
(B) 6
(C) 7
(D) 9
(E) 11

Let me know if there are multiple ways to solve this type of questions?

[Ian Stewart]

How many different positive integers are factors of 441?

There are different ways to answer the question- you could try to list all of the factors, for example. It's always easiest to see factors, though, if you have a prime factorization. 441 is likely an unfamiliar number, but you can see straight away that you can divide by 9 (4+4+1 = 9)

441 = 9*49 = 3^2 * 7^2

Now you can see all nine of the factors:
1
3
7
3*7
3^2
7^2
3 * 7^2
3^2 * 7
3^2 * 7^2

Faster still, if you have the prime factorization, look only at the powers. Add one to each power, and multiply- that will give you the total number of positive divisors. In this case we have the prime factorization 3^2 * 7^2. Add one to each power- we get 2+1 = 3 and 2+1 = 3; multiply to get 3*3 = 9 positive divisors of 441.