In a two-digit number, the product of the digits is 8. When 18 is added to the…

Concept

For any two-digit number, write the tens digit as t and the units digit as u, so the number equals 10t + u and the digits-reversed number equals 10u + t. A digit-product condition (tu = k) combined with a reversal condition (formed by adding/subtracting a constant to reverse the digits) gives two equations in t and u; these can be solved by direct substitution, and cross-checked independently using the identity (t + u)2 = (t - u)2 + 4tu, which relates the sum and difference of two numbers to their product.

Step-by-step application

1. Let the tens digit be t and the units digit be u, so the number is 10t + u and the reversed number is 10u + t.

2. The product of the digits is 8, so tu = 8.

3. Adding 18 to the number reverses its digits: 10t + u + 18 = 10u + t.

4. Simplify: 9t − 9u = −18, so t − u = −2, i.e., u = t + 2.

5. Substitute into tu = 8: t(t + 2) = 8, i.e., t2 + 2t − 8 = 0, which factors as (t + 4)(t − 2) = 0.

6. Since a digit cannot be negative, reject t = −4 and take t = 2, giving u = 4.

7. The number is 10t + u = 10 × 2 + 4 = 24.

Cross-check

Independent check via the sum–difference identity: (t + u)2 = (t − u)2 + 4tu = (−2)2 + 4×8 = 36, so t + u = 6; combined with t − u = −2 this again gives t = 2, u = 4 — confirming the number 24. Also, plugging back into the original conditions: 2 × 4 = 8 (product matches) and 24 + 18 = 42, the reverse of 24 (reversal matches).