A nursery worker has 72 small trees to be displayed in lines in a rectangular shape. Each horizontal and vertical line must have more than 5 trees. How many different arrangements are possible?

1 Answer
Feb 26, 2018

2 arrangements
6*12
8*9

Explanation:

The important part of this question is to recognize factor pairs of 72 that are both larger than 5.

By prime factorization, we have
72=2^3*3^2,
Which gives a total of (3+1)(2+1)=12 factors, and hence 6 pairs of factors.

Note that 1, 2, 3, 4 (which is 2*2) are smaller than 5, so we have to deduct 4 pairs, giving the answer of 2 pairs.

Method 2: Listing

72
=1*72
=2*36
=3*24
=4*18
=6*12
=8*9

As we can see, there are only 6*12 and 8*9 that satisfy the requirement.