It isn't divisible

Number Theory Level 5

A and B are distinct two-digit positive integers. The four-digit number X is formed by writing down the number B followed by the number A, and the four-digit number Y is formed by writing down the number A followed by the number B.

What is the smallest positive integer that does not divide $|X-Y|$ for any values of A and B?

The answer is 91.

1 solution

Anirudh Sreekumar
Mar 19, 2017

The 2 digit numbers are $A,B$

The 4 digit numbers $\overline{AB}$ and $\overline{BA}$ can be witten as,

$\overline{AB}=100A+B$

$\overline{BA}=100B+A$

we get,

$|\overline{AB}-\overline{BA}|=|(100A+B)-(100B+A)|=99|A-B|$

now $|A-B|$ can take values from $1$ to $89$ (as $A$ and $B$ are $2$ digit numbers)

also if $|A-B|=10,30$ ,

$99|A-B|$ is divisible by $90$

$91=13\times7$ is coprime to $99$ and thus is the smallest number that doesn't divide $|\overline{AB}-\overline{BA}|$ for any $A,B$

It isn't divisible

The answer is 91.

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