flying acrobats raring to go!

ashish1354 wrote:The flying acrobatic team is made up of 120 airplanes. The team wants to form a rectangular formation with X planes in a row and Y planes in a column. If the number of airplanes in a row is no less than 4 and no more than 30, how many different combinations of rectangular shapes are possible?

(a) 4.
(b) 5.
(c) 6.
(d) 8.
(e) 10.

can someone show a fast way to solve this

For the formation to be a rectangle, we can see that X and Y must be factors (divisors) of 120.
Once we select the value of X (# of planes in a row), the value of Y follows (e.g., if X=20, then Y must equal 6).
So, how many values of X can we have?
Well, let's list all of the factors of 120.
120: {1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120}
Given the conditions for the values of X in the question, we see that X can equal 4,5,6,8,10,12,15,20,24,and 30
So, the answer is E

As far as a "quick" way to find divisors is concerned, we could use a nice rule involving the prime factorization (see below), but the restrictions regarding the value of X prevent this from being as useful as usual. For this question, listing the divisors might be the fastest way.

Aside:
Finding the total number of divisors of positive integer N:
- Write the prime factorization of N
- If the prime factorization of N = (W^a)(X^b)(Y^c). . ., then the total number of divisors of N=(a+1)(b+1)(c+1)...