Question

prove that if an even integer n is subtracted from an odd integer m. then m - n is odd.

Answer 1

Proof :

We know that an even integer is the one which is exactly divisible by 2 i.e., an even integer is always a multiple of 2.

Let n be an even integer , then n can also be written as the multiple of 2 .

i.e., n = 2h .........(1)

where h is some other integer .

Also let m be an odd integer , then m can also be written as -

m = 2k+1 ...........(2)

where k is some other integer .

Now , subtracting equation (1) from equation (2) , we get -

Put (k-h) = u .

Here , u will also be an integer because difference of two integers is always an integer (i.e., integers are closed under subtraction )

i.e., we have -

, u is an integer .

Now , since u is any integer , then 2u , being a multiple of 2 , will be even and hence (2u+1) will be odd.

Thus , from above it is clear that (m-n) is odd .

Hence , we get -

The difference (m-n) of an odd integer m and an even integer n , is odd .

Proved !