Question

When a two-digit number is divided by the sum of its digits, the quotient is 7 and the remainder is 6. If one of the digits of the number is three, then what is the difference of the digits?

• A. 2
• B. 3
• C. 4
• D. 5
• E. 6

Answer 1

The Correct Option is D

To solve this problem, we need to understand the conditions given: 

1. We are dealing with a two-digit number.
2. This number, when divided by the sum of its digits, gives a quotient of 7 and a remainder of 6.
3. One of the digits of the number is 3.

Let the two-digit number be represented as 10a + b, where a and b are the digits of the number. The sum of its digits is a + b.

According to the problem:

\(\frac{{10a + b}}{{a + b}} = 7\ \text{(quotient)} + \frac{6}{{a + b}}\ \text{(remainder)}\)

Rewriting the equation, we have:

\(10a + b = 7(a + b) + 6\)

Expand and rearrange the terms:

\(10a + b = 7a + 7b + 6\)

Simplifying, we get:

\(3a - 6b = 6\)

Further simplifying, we divide the entire equation by 3:

\(a - 2b = 2\)

Given that one of the digits is 3, let's consider "b = 3" as our assumption:

Substitute \(b = 3\) into the equation \(a - 2b = 2\):

\(a - 2(3) = 2\)

This simplifies to:

\(a - 6 = 2\)

\(a = 8\)

Thus, the number is 10a + b = 10(8) + 3 = 83. We verify it by checking:

Sum of digits of 83 is \(8 + 3 = 11\).

Quotient check: \(\frac{83}{11} = 7\ \text{remainder}\ 6\) which holds true.

Therefore, the difference between the digits is:

\(|8 - 3| = 5\)

Hence, the difference of the digits is 5, which is the correct answer.