Question

The product of the digits of a three-digit number is 70. The sum of the digits of the three-digit number is:

• A. 16
• B. 14
• C. 12
• D. 18

Hint:

For problems involving the product of digits, prime factorization is the most effective starting point. It breaks down the product into its fundamental building blocks, which are often the digits themselves or can be combined to form the digits.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are looking for a three-digit number where the product of its three digits equals 70. After finding the digits, we need to calculate their sum.
Step 2: Key Formula or Approach:
The best approach is to find the prime factorization of the product, 70. The factors will represent the digits of the number.
Step 3: Detailed Explanation:
Let the three digits of the number be x, y, and z. The digits must be integers from 0 to 9.
We are given that their product is 70:
\[ x \times y \times z = 70 \] First, find the prime factorization of 70.
\[ 70 = 2 \times 35 = 2 \times 5 \times 7 \] The prime factors are 2, 5, and 7. All three of these are single-digit numbers.
This is the only combination of three distinct single-digit numbers (other than using 1, which would require a two-digit factor like 10 or 14) that multiply to 70.
So, the three digits of the number must be 2, 5, and 7.
The possible three-digit numbers are permutations of these digits (e.g., 257, 275, 527, etc.), but the digits themselves are fixed.
The question asks for the sum of the digits.
\[ \text{Sum of digits} = 2 + 5 + 7 \] \[ \text{Sum of digits} = 14 \] Step 4: Final Answer:
The sum of the digits of the three-digit number is 14.