Question

Mrs. Weber, the math teacher, has 45 logic puzzles and 10 visual puzzles that she wants to group into sets for students who finish their tests early. Mrs. Weber wants each set to be identical, containing the same combination of logic puzzles and visual puzzles, with no puzzles left over. What is the greatest number of sets she can create?

Answer 1

To find the greatest number of identical sets that Mrs. Weber can create from the 45 logic puzzles and 10 visual puzzles, we need to determine the greatest common divisor (GCD) of these two numbers.

We will start by finding the prime factorization of both numbers.

**Step 1: Prime Factorization**

For 45:
- \( 45 = 5 \times 9 = 5 \times 3^2 \)

Thus, the prime factorization of 45 is:
\[
45 = 3^2 \times 5^1
\]

For 10:
- \( 10 = 2 \times 5 \)

Thus, the prime factorization of 10 is:
\[
10 = 2^1 \times 5^1
\]

**Step 2: Find the GCD**

To find the GCD, we identify the minimum power for each prime factor present in both factorizations.

- The prime factor \( 2 \) appears in \( 10 \) but not in \( 45 \) (minimum power = 0).
- The prime factor \( 3 \) appears in \( 45 \) but not in \( 10 \) (minimum power = 0).
- The prime factor \( 5 \) appears in both: the minimum power of \( 5 \) is \( 1 \).

Thus, the GCD is:
\[
GCD = 2^0 \times 3^0 \times 5^1 = 1 \times 1 \times 5 = 5
\]

**Step 3: Conclusion**

The greatest number of identical sets Mrs. Weber can create, containing the same combination of logic puzzles and visual puzzles with no puzzles left over, is:
\[
\boxed{5}
\]