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Problem 267

Number theory Difficulty 2.3 Find the answer CEMC Fermat · Canada · 2021

The digits in a two-digit positive integer are reversed. The new two-digit integer minus the original integer equals 54. What is the positive difference between the two digits of the original integer?

$5$
$7$
$6$
$8$
$9$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Suppose that the original integer has tens digit $a$ and ones (units) digit $b$ .

This integer is equal to $10a+b$ .

When the digits are reversed, the tens digit of the new integer is $b$ and the ones digit is $a$ .

This new integer is equal to $10b+a$ .

Since the new two-digit integer minus the original integer is 54, then $(10b+a)-(10a+b)=54$ and so $9b-9a=54$ which gives $b-a=6$ .

Thus, the positive difference between the two digits of the original integer is 6. An example of a pair of such integers is 71 and 17.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.