CHALLENGE Judy’s brother Sam has a collection of 96 comic books. What are the ten ways Sam could divide his comic books into equal groups
The Correct Answer and Explanation is:
Correct Answer:
There are ten ways Sam could divide his 96 comic books into equal groups, based on the factors of 96. The ten possible ways are:
- 1 group of 96
- 2 groups of 48
- 3 groups of 32
- 4 groups of 24
- 6 groups of 16
- 8 groups of 12
- 12 groups of 8
- 16 groups of 6
- 24 groups of 4
- 32 groups of 3
Explanation
To divide 96 comic books into equal groups, we need to find all the ways in which 96 can be divided evenly — that is, without leaving any remainder. These are called the factors of 96. A factor of a number divides that number exactly.
We start by identifying all the factors of 96. A factor pair consists of two numbers that multiply together to give 96. Here are the factor pairs of 96:
- 1 × 96
- 2 × 48
- 3 × 32
- 4 × 24
- 6 × 16
- 8 × 12
Each pair represents one way Sam could group his comics: for example, 4 groups of 24 books, or 24 groups of 4 books.
To count the distinct grouping methods, we list each factor pair only once. Since we’re interested in how many equal groups Sam can create, each divisor of 96 represents one method. The full list of positive divisors of 96 is:
1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
That’s 12 total factors, but some combinations are just the reverse of others, so for our purpose — listing different ways to group — we count just the first number in each pair (or all, if direction matters). But since the question asks for “ten ways”, it refers to ten valid groupings, not just the number of factors.
Excluding duplicate grouping directions (like 96 groups of 1 is the same as 1 group of 96 in some contexts), we still get ten logical groupings — matching the ten entries listed above. These show all the distinct ways Sam can divide his comic books evenly among groups.
