We want to calculate the greatest number of friends we can invite to an arcade using the given coupon such that the tokens and slices of pizza are equally split between we and our friends with none left over.
There are 24 slices of pizza. This means that we need to find two numbers that give us 24 when we multiply them. One of the numbers will be the slices of pizza and the other number the number of friends. Let's start by listing the factor pairs of 24.
24 &= 1 * 24
24 &= 2 * 12
24 &= 3 * 8
24 &= 4 * 6
There are 8 possible arrangements for slices of pizza.
1 slice of pizza for 24 people
2 slices of pizza for 12 people
3 slices of pizza for 8 people
4 slices of pizza for 6 people
24 slices of pizza for 1 people
12 slices of pizza for 2 people
8 slices of pizza for 3 people
6 slices of pizza for 4 people
Now let's list the factor pairs of 90 to know how to split the tokens.
90 &= 1 * 90
90 &= 2 * 45
90 &= 3 * 30
90 &= 5 * 18
90 &= 6 * 15
90 &= 9 * 10
This means that there are 12 arrangements for tokens.
1 token with 90 people
2 tokens for 45 people
3 tokens for 30 people
5 tokens for 18 people
6 tokens for 15 people
9 tokens for 10 people
90 tokens for 1 people
45 tokens for 2 people
30 tokens for 3 people
18 tokens for 5 people
15 tokens for 6 people
10 tokens for 9 people
The numbers that form the factor pairs are the factors of the numbers.
Factors of24 &: 1,2,3, 6,8,12,24
Factors of90 &: 1,2,3,5, 6,9,10,15,18,...
Notice that the greatest common factor between the numbers is 6. This means that the greatest number of people that we can split the coupon with is 6. Then, the possible arrangements will be only the ones with 6.
4 slices of pizza for 6 people
15 tokens for 6 people
Since we are including ourselves in the group, we can invite 6-1=5 friends to the arcade with us. Each of us will get 15 tokens and 4 slices of pizza.
