Lee is sewing vest using 16 green buttons and 56 blue buttons. all the vests are identical and all have both green and blue buttons. what are the possible numbers of vests lee can make? what is the greatest number of vests lee can make.

For all the answers I will assume that all buttons both green and blue must be used. 

For the first question we will start by searching all the divisors of 16;

16: 1, 2, 4, 8, 16
We will think about the numbers of green buttons that can be used as the divisors of 16. We will check if the dividend of 16 to its divisors is a divisor for 56.

1 => 16 vests with 1 green button 56/16 = 3.5
2=> 8 vests with 2 green buttons 56/8 = 7
4=>  4 vests with 4 green buttons 56/4 = 14
8=> 2 vests with 8 green buttons 56/2 = 28
16=> 1 vest with 16 green buttons 56/1 = 56

From all the above results we concluded that Lee can make:
8 vests with 2 green buttons and 7 blue buttons
4 vests with 4 green buttons and 14 blue buttons
2 vests with 8 green buttons and 28 blue buttons
1 vest with 16 green buttons and 56 blue buttons

For the second question, the greatest number of vests represents the greatest common divider of 16 and 56. 
First, we break into prime factors
16 = 2^4
56 = 2^3 *7
The greatest common divider is 8.
Lee can make 8 vests and each one will have 2 green buttons and 7 blue buttons (16/8=2 and 56/8=7)