Divisibility by 4

Rule for Divisibility by 4

A number is divisible by 4 if the last two digits of the number are divisible by 4 .

Examples

A.)2,234,489,032 is divisible by 4 because its last two digits, 32, form a number divisible by 4.

B.) 789,079 is not divisible by 4 because its last three digits, 79, form a number not divisible by 4.

Proof

For any integer x written as a_na_n-1a_n-2...a₁a₀, we will show that x is divisible by 4 if a₁a₀ is divisible by 4.

If we write x as a_na_n-1a_n-2...a₁a₀, then we can also write:

x= a₀ + a₁(10) + a₂(10² )+ a₃(10³)... + a_n-2(10^n-2) + a_n-1(10^n-1 ) + a_n(10ⁿ)

= (a_n×10ⁿ + a_n-1×10^n-1 + a_n-2×10^n-2 + .... + a₂)×100 + a₁a₀

= 4×25× (a_n×10ⁿ + a_n-1×10^n-1 + a_n-2×10^n-2 + .... + a₂) + a₁a₀

Since the term 4×25×(a_n×10ⁿ + a_n-1×10^n-1 + a_n-2×10^n-2 + .... + a₂) is divisible by 4, the integer a_na_n-1a_n-2 ....a₁a₀ is divisible by 4 if and only if the number a₁a₀ is divisible by 4 and vice versa.

Below is an alternate proof using modular arithmetic.