Question

The product of a set of distinct positive integers is $630$.  If one of the numbers is 42, what are the other two numbers?

Answer 1

To determine the other two numbers in the set of distinct positive integers that multiply to 630, given that one of the numbers is 42, we start by factoring 630:

1. **Prime Factorization of 630:**
- First, divide by 2: \( 630 \div 2 = 315 \)
- Then, divide by 3: \( 315 \div 3 = 105 \)
- Again, divide by 3: \( 105 \div 3 = 35 \)
- Next, divide by 5: \( 35 \div 5 = 7 \)

Thus, the prime factorization of \( 630 \) is:
\[
630 = 2 \times 3^2 \times 5 \times 7
\]

2. **Evaluate and Confirm 42's Factors:**
- \( 42 = 2 \times 3 \times 7 \)

3. **Determine the Product of Remaining Integers:**
Given one number is 42, calculate the quotient of \( 630 \div 42 \) to find the product of the other two numbers:
\[
\frac{630}{42} = 15
\]

4. **Factorize 15:**
- The factors of \( 15 \) are \( 3 \times 5 \).

Thus, if one of the numbers is 42, the other two numbers must be 3 and 5, since:

- \( 42 \times 3 \times 5 = 630 \)

Therefore, the set of distinct positive integers is \( \{3, 5, 42\} \).

To verify, re-expand the verification steps:
\[
42 \times 3 = 126, \quad 126 \times 5 = 630
\]

Thus, the other two numbers are \( \boxed{3 \text{ and } 5} \).