#### Answer

The men and women should be sorted into groups of 36.

#### Work Step by Step

180 men and 144 women are in a choir. The director needs to divide the larger groups into smaller, equal-sized groups. The director wants each group to be the same size and only consist of men or women.
If we find the Greatest Common Divisor (GCD) of 144 and 180, we will have the smallest number that divides evenly (without a remainder) for 144 and 180. This will give us the smallest sized groups of equal numbers.
To find the GCD of 180 and 144, we can prime factor both numbers and then choose the factors common to both numbers.
Prime factoring:
180 = 2 x 2 x 3 x 3 x 5
144 = 2 x 2 x 2 x 2 x 3 x 3
The common prime factors are: 2 x 2 x 3 x 3 = 36.
Therefore, the director can make 4 groups of women (with 36 women in each group) and 5 groups of men (with 36 men in each group).