# Palindromes and LCM

A palindrome is a number that is the same when read forwards and backwards, such as $43234$. What is the smallest five-digit palindrome that is divisible by $11$?

I'm probably terrible at math but all I could do was list the multiples out. Any hints for a quicker solution

$$\overline{abcba}\equiv 10^4\cdot a+10^3\cdot b+10^2\cdot c+10^1\cdot b+a$$

$$\equiv (-1)^4a+(-1)^3b+(-1)^2c+(-1)^1b+a$$

$$\equiv a-b+c-b+a\equiv 2a-2b+c\equiv 0\pmod{11}$$

To minimize $\overline{abcba}$, let $a=1$ and $b=0$. Then $c\equiv 9\pmod{11}$, so $c=9$. And in fact $\overline{10901}$ works.

• I'm terribly confused. Did you mess up latex?? – Houdineo Oct 4 '15 at 21:42
• What is mod? Is it modular stuff? I'm not that quite advanced in math. – Houdineo Oct 4 '15 at 21:45
• Im stating the obvious but this is assuming that $a$ can't be zero. Maybe thats part of the definition of palindrome, Im not sure – Elliot G Oct 4 '15 at 21:45
• @Houdineo $\overline{abcba}$ denotes the integer with digits $a,b,c,b,a$. Also $\equiv$ denotes equivalence/congruence mod $11$. See Wikipedia. – user236182 Oct 4 '15 at 21:45
• But I love when proofs like this end up being so simple – Elliot G Oct 4 '15 at 21:46