Divisibility by 25

Rule for Divisibility by 25

A number is divisible by 25 if it ends with 00, 25, 50, or 75.

Examples

A.) 154,750 is divisible by 25 because its ends with 50.

B.) 6,783,034 is not divisible by 25 because is does not end with 00, 25, 50, or 75.

Proof

For any integer a_na_n-1a_n-2...a₁a₀, we will show that x is divisible by 25 if a₁a₀ is divisible by 25.

If we write x as a_na_n-1a_n-2...a₂a₁a₀, 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

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

Since the term 25×4×(a_n×10ⁿ + a_n-1×10^n-1 + a_n-2×10^n-2 + .... + a₂) is divisible by 25, the integer a_na_n-1a_n-2 ....a₁a₀ is divisible by 25 if and only if the number a₁a₀ is divisible by 25 and vice versa. The only four combinations of last digits divisible by 25 are 00, 25, 50, and 75 [25=1×25, 50=2×25, 75=3×25, (1,2,3,4,5,6,7,8, or 9)00=(1,2,3,4,5,6,7,8 or 9)×4×25].