Question

If m and n are odd positive integers, then m^2 + n^2 is even, but not divisible by 4. Justify.

Answer 1

Step-by-step explanation:

Given:-

m and n are odd positive integers

To find:-

If m and n are odd positive integers, then m^2 + n^2 is even, but not divisible by 4.

Solution:-

Given that:-

m and n are odd positive integers

we know that

the general form of an odd positive integer is 2a+1

now

let take m = 2a+1 and n=2b+1

Now

m^2 = (2a+1)^2

It is in the form of (x+y)^2

Where ,x= 2a and y=1

we know that (x+y)^2 = x^2 +2xy +y^2

=>m^2 = (2a)^2+2(2a)(1)+(1)^2

m^2 = 4a^2 +4a + 1 -------------------------(1)

and

n^2 = (2b+1)^2

It is in the form of (x+y)^2

Where ,x= 2b and y=1

we know that (x+y)^2 = x^2 +2xy +y^2

=>n^2 = (2b)^2+2(2b)(1)+(1)^2

n^2 = 4b^2+4b+1 ---------------------------(2)

On adding (1)&(2) then

m^2+n^2 = (4a^2+4a+1)+(4a^2+4a+1)

=>m^2+n^2 =4a^2+4a+1+4b^2+4b+1

=>m^2+n^2 = 4a^2+4a+4b+4b^2+2

=>m^2+n^2=2(2a^2+2a+2b+1)

=>m^2+n^2 =2c

Where 2a^2+2a+2b+1 =c

RHS is in the general form of an even number

So,m^2+n^2 is an even number -------(3)

and

=>m^2+n^2=4(a^2+a+b+b^2)+2

=>m^2+n^2 =4d+2

Where d =a^2+a+b+b^2

It is not divisible by 4-----------(4)

(it leaves the remainder 2)

From (3)&(4)

m^2+ n^2 is even, but not divisible by 4

Answer:-

m and n are odd positive integers, then m^2 + n^2 is even, but not divisible by 4.

Check :-

Let consider any two odd positive integers

3 and 5

Let m=3 and n=5

m^2 = 3^2 =9

n^2 = 5^2 =25

m^2+n^2 = 9+25 =34

34 is an even and but not divisible by 4

Hence , Verified

Used Concept:-

• The general form of an even number = 2m
• If m=1,2,3... then we get 2,4,6,8....
• The general form of an odd number= 2m+1
• If m=1,2,3... then we get 3,5,7,9,...