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?

Question

Arun selected an integer \( x \) between 2 and 40, both inclusive. He noticed that the greatest common divisor of the selected integer \( x \) and any other integer between 2 and 40, both inclusive, is 1. How many different choices for such an \( x \) are possible?

Answer 1

The correct answer is option (D):

5

Let the two-digit number be represented as 10a + b, where 'a' is the tens digit and 'b' is the units digit. The problem states that when this number is divided by the sum of its digits (a + b), the quotient is 7 and the remainder is 6. This can be expressed as:

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

Simplifying this equation:

10a + b = 7a + 7b + 6
3a - 6b = 6
a - 2b = 2 (Equation 1)

We are also given that one of the digits is 3. We can consider two cases:

Case 1: a = 3
Substitute a = 3 into Equation 1:
3 - 2b = 2
-2b = -1
b = 1/2
Since 'b' must be an integer, this case is not valid.

Case 2: b = 3
Substitute b = 3 into Equation 1:
a - 2(3) = 2
a - 6 = 2
a = 8
So the two-digit number is 83. The sum of the digits is 8 + 3 = 11. When we divide 83 by 11, we get a quotient of 7 and a remainder of 6, as required (83 = 7 * 11 + 6).

The digits are 8 and 3. The difference between the digits is |8 - 3| = 5. Therefore, the correct answer is 5.

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?

The Correct Option is D

Solution and Explanation

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