Question

Prove that a 4-digit palindrome is always divisibdivisibl 11

Answer 1

Suppose a and b are digits, and we consider the palindrome that looks like abba.

Now, abba = 1000*a + 100*b + 10*b + a.

We can rearrange this to
1000*a + a + 100*b + 10*b = 1001*a + 110*b.

Now, 1001 = 7 * 11 * 13, and 110 = 2 * 5 * 11,

so, abba = 1001*a + 110*b = 11 * (7*13*a + 2*5*b)

So abba is a multiple of 11.

Here's a specific example:

3223 = 3000+200+20+3 = 3003+220
= 11*273+11*20 = 11*293


So it is always divisible by 11