Question

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?

Answer 1

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.