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

The correct option is 14.

To find the sum of the digits of the three-digit number, we can break down the problem by analyzing the digits and their product.

Let the three digits of the three-digit number be represented by the variables $a$, $b$, and $c$. Since these are digits of a number and their product is non-zero, each digit must be a single-digit integer from 1 to 9:

We are given that the product of these three digits is 70:
$a\times b\times c=70$

To determine the possible values for $a$, $b$, and $c$, we find the prime factorization of 70:
$70=2\times 5\times 7$

The prime factors of 70 are 2, 5, and 7. Since 5 and 7 are prime numbers and must be factors of the digits, and the only single-digit multiples of 5 and 7 are 5 and 7 themselves, two of our digits must be 5 and 7. The third digit must therefore be 2.

Thus, the three digits of the number are 2, 5, and 7 (in any order, such as 257, 572, 752, etc.).

Now, we calculate the sum of these digits:
$\text{Sum of digits}=2+5+7=14$