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 anan-1an-2...a1a0, we will show that x is divisible by 25 if a1a0 is divisible by 25.
If we write x as anan-1an-2...a2a1a0, then we can also write:
x= a0 + a1(10) + a2(102 )+ a3(103)... + an-2(10n-2) + an-1(10n-1 ) + an(10n)
= (an×10n + an-1×10n-1 + an-2×10n-2 + .... + a2)×100 + a1a0
= 25×4× (an×10n + an-1×10n-1 + an-2×10n-2 + .... + a2) + a1a0
Since the term 25×4×(an×10n + an-1×10n-1 + an-2×10n-2 + .... + a2) is divisible by 25, the integer anan-1an-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].