Question

The sum of the digits of a two-digit number is 10. If 18 is subtracted from it, the digits in the resulting number will be equal. The number is:

• A. 73
• B. 75
• C. 65
• D. 64

Answer 1

The Correct Option is A

To solve the problem, we need to find a two-digit number based on the following conditions: the sum of its digits is 10, and after subtracting 18, the resulting number has equal digits. 

Let the two-digit number be represented as \(10a + b\), where \(a\) is the tens digit and \(b\) is the units digit. The conditions can be expressed as:

1. \(a + b = 10\)

2. \(10a + b - 18\) results in a number with equal digits, say \(11c\). Therefore, \(10a + b - 18 = 11c\).

By rearranging the second equation, we have:

\(10a + b = 11c + 18\).

Since the result \(11c\) must be a two-digit number with equal digits, \(c\) can only be 1 to 9. Let's start by solving for \(c\).

Substituting the sum condition \(a + b = 10\) into the equation obtained, we organize as:

\(b = 10 - a\)

Substitute \(b\) into \(10a + b = 11c + 18\):

\(10a + (10 - a) = 11c + 18\)

\(9a + 10 = 11c + 18\)

\(9a = 11c + 8\)

Testing valid values for \(c\), we try to find integer solutions for \(a\) and \(b\):

When \(c = 5\), \((11 \cdot 5) + 8 = 55 + 8 = 63\), thus:

\(9a = 63\)

\(a = 7\)

Then \(b = 10 - a = 10 - 7 = 3\)

Thus the calculated number is: \(10a + b= 10\cdot7 + 3 = 73\).

Hence, the correct number is 73.