# Divisibility by 4

1. If the number formed by the two digits at the tens place and unit place is divisible by 4 then the number is divisible by 4.

2. If two times the digit at the tens place when added to the digit at the unit place is divisible by 4 then the number is divisible by 4 else not.

## Proof, How: Division by four

Let the number be abcd.

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

which can also be written as
a×4×250 + b×4×25 + c×10 + d

Every power of 10 i.e. 10x is divisible by 4 if x>2.

So a×4×250 + b×4×25 is divisible by 4 as it has 4 as factor or common factor.

The number is divisible by 4 if 10×c+d or cd is divisible by 4.

The digits left is c×10+d

Which can be written as
c(4×2+2)+ d = c×4×2 + 2c + d

The above is divisible by 4 if 2c+d is divisible by 4.

## Remainder of division by 4

1. The remainder of the number formed by the last two digits when divided by 4 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 2a+b, when divided by 4.

## Examples of division by 4

1. Is 348 divisible by 4.
The number formed by the last two digits is 48. 48 is divisible by 4 therefore the number is divisible by 4.

2. Is 3483 divisible by 4.
The number formed by the last two digits is 83. 83 is not divisible by 4 therefore the number is not divisible by 4. The remainder is the remainder of 83 when divided by 4. The remainder is 3.

3. Is 375 divisible by 4.
The number formed by the product of digit at tens place and digit at unit place added to it is (7×2+5)=19. The remainder when 19 is divided by 4 is 3. Hence the remainder is 3.