$XY$ minus $YX$ is always divisible by $3$

Question

Why a $2$-digit decimal number minus the same number with the digits reversed is always divisible by $3$ ?

hint: what is the digit sum of the difference? – Bananach Mar 01 '17 at 08:31 — Bananach, Mar 01 '17 at 08:31
Hint: instead of XY write $10x+y$. – Carsten S Mar 01 '17 at 08:31 — Carsten S, Mar 01 '17 at 08:31

score 5 · Accepted Answer · answered Mar 01 '17 at 08:34

5

We write: $XY=10X+Y, YX=10Y+X$. Hence

$XY-YX=10(X-Y)-(X-Y)=9(X-Y)$.

Your turn !

answered Mar 01 '17 at 08:34

Fred

score 4 · Answer 2 · answered Mar 01 '17 at 08:37

Note that $`XY\text{'}$ is convenient shorthand for $10X + Y$. For example, $93$ is shorthand for $10 \cdot 9 + 3$.

So $$ `XY\text{'} - `YX\text{'} = (10 X + Y) - (10Y + X) = 9X - 9Y = 9(X - Y), $$ which is divisible by $9$ and in particular by $3$.

A great example of the power of simple algebra :)

2 Answers2