Question

If \( xyz \) is a three-digit number and the product of its digits is \( 70 \), then find the sum of the digits.

• A. 16
• B. 14
• C. 18
• D. 12

Hint:

When product of digits is given, first prime-factorize the number and check which factors are valid digits (1–9).

Answer 1

The Correct Option is A

Step 1: Understanding the Topic
This is a basic number theory problem. We are given the product of the three digits (x, y, z) of a number and asked to find their sum. The key constraints are that x, y, and z must be single-digit integers, and since it's a three-digit number, the first digit x cannot be zero.
Step 2: Key Approach - Prime Factorization
The most direct way to solve this is to find the prime factorization of the product, 70. This will give us the fundamental building blocks of the three digits.
Step 3: Detailed Calculation
A. Find the prime factorization of 70:
\[ 70 = 10 \times 7 = (2 \times 5) \times 7 \] The prime factors are 2, 5, and 7.
B. Determine the three digits:
We need to find three single-digit integers whose product is 70. The prime factors themselves (2, 5, and 7) are all single-digit integers. Therefore, the only possible combination of three digits that multiply to 70 is the set $\{2, 5, 7\}$. Any other combination, like $\{1, 7, 10\}$, is invalid because 10 is not a single digit.
So, the digits x, y, and z are 2, 5, and 7 in some order.
C. Calculate the sum of the digits:
The sum of the digits is independent of their order. \[ \text{Sum} = 2 + 5 + 7 = 14 \] Step 4: Final Answer
The sum of the digits is 14. \[ \boxed{14} \]