# Are there two integers $x,y>1$ such that $x\times y$ is the number you get when reversing the digits of $x$?

Title pretty much says it all.

For example, if $x=121149$, one needs to find an integer $y$ such that $x\times y = 941121$.

• I'm guessing you don't want the "x any palindromic number; y = 1" answer? – EagleV_Attnam May 4 '15 at 8:32
• @EagleV_Attnam - Indeed, I will edit the question, thanks. – R B May 4 '15 at 8:33
• I suspect that the only numbers that have this property are numbers with a trailing zero, eg. $440 = 044 \times 10$. However, I'm not smart enough to prove it :P – Tryth May 4 '15 at 8:42
• @Tryth Check my answer! :-) I was trying to prove it impossible, but eventually restricted the cases enough that I managed to find an actual example. – Rand al'Thor May 4 '15 at 8:44
• @randal'thor Nice find! – Tryth May 4 '15 at 8:46

Cheat example:

$x=11,y=1,x\times y=11$.

More serious example:

$x=2178,y=4,x\times y=8712$.

Some theory:

assume $x,y$ are two integers as stated. Then $x$ and $x\times y$ have the same digit sum, so they're congruent mod $9$. So $x(y-1)$ is a multiple of $9$. But $y$ must be less than $10$, so $y-1$ can't be a multiple of $9$. So either $x$ is a multiple of $9$ or $x$ and $y-1$ are both multiples of $3$. In the latter case, $y$ must be either $4$ or $7$.

• @Downvoter, I edited the question to x,y>1 only after this was posted, please revert your vote. – R B May 4 '15 at 8:35
• @RB I've solved it properly now! – Rand al'Thor May 4 '15 at 8:44
• Nice example. Did you find it using a computer or hand crafted :) ? – R B May 4 '15 at 8:47
• @RB Using a computer, sadly :-( – Rand al'Thor May 4 '15 at 8:49
• OEIS has a sequence for everything. – Tryth May 4 '15 at 8:50