25

Divisibility by 25

  1. If the number formed by the two digits at the tens place and unit place in order is divisible by 25 then the number is divisible by 25 else not.

  2. If ten times the digit at the tens place when added to the digit at the unit place is divisible by 25 then the number is divisible by 25 else not. The digit at the unit place must be zero or 5.

Proof:

Let the number be abcd.

abcd can be written as
a×1000 + b×100 + c×10 + d

which can also be written as
a×25×40 + b×25×4 + c×10 + d

Every power of 10 i.e. 10x is divisible by 25 if x>1.

So a×25×40 + b×25×4 is divisible by 25 as it has 25 as factor or common factor.

The number is divisible by 25 if  cd is divisible by 25.

The digits left is c×10+d
d can not be other than 0 or 5, the digit at tens place can be 0,2,5 and 7.

The above is divisible by 25 if 10c+d is divisible by 25.
Remainder

  1. The remainder of the number formed by the last two digits when divided by 25 is the remainder of the original number.

  2. If the number formed by the last two digits is ab. Then the remainder is the remainder of the number 10a+b, when divided by 25.

Examples

  1. Is 358 divisible by 25.
    The number formed by the last two digits is 58. 58 is not divisible by 25 it is greater by 8 therefore the number is not divisible by 25. Remainder is 8.

  2. Is 3450 divisible by 25.
    The number formed by the last two digits is 50. 50 is divisible by 25 therefore the number is divisible by 25.


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