Divisibility by 11
Rule for Divisibility by 11
A number is divisible by 11 if the alternating sum of its digits is divisible by 11. So if anan-1an-2...a2a1a0 is divisible by 11, then an - an-1+ an-2 .... + a2(-1)n-2 + a1(-1)n-1 + a0(-1)n is also divisible by 11 and vice versa.
Examples
A.) 280,819: 2-8+0-8+1-9=-22, so it is divisible by 11 (recall the definition of divisbility allows for negative numbers).
B.) 53: 5-3=2, so it is not divisible by 11.
Proof
This rule will be proved using modular arithmetic and the proof helps to illustrate the power of modular arithmetic.
Notice that 10≡-1 (mod 11). Therefore 10k ≡ (-1)k (mod 11) for k=1 ,2 ,3, 4…. Then
x≡ a0 + a1(-1)+ a2 (-1)2 + a3(-1)3+ …+ am (-1)m (mod 11)
≡ a0 - a1+ a2 - a3+ …+ am (-1)m (mod 11)
Therefore x is divisible by 11 if and only if the alternating sum of its digits, an - an-1+ an-2 .... + a2(-1)n-2 + a1(-1)n-1 + a0(-1)n , is divisible by 11.