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Prove that if x and y are both odd positive integers, then x² + y²is even but not divisible by 4

Solution:

Consider the two odd positive numbers x and y to be 2k + 1 and 2p + 1.

We can write it as

x²+ y² = (2k + 1)² + (2p + 1)²

Using the algebraic identity (a + b)² = a² + b²+ 2ab

= 4k² + 4k + 1 + 4p² + 4p + 1

On rearranging

= 4k² + 4p² + 4k + 4p + 2

Taking out 4 as common

= 4(k² + p² + k + p) + 2.

Thus, the sum of squares is even if the number is not divisible by 4.

Therefore, if x and y are odd positive integers, then x² + y² is even but not divisible by 4.

✦ Try This: Prove that if x and y are both odd positive integers, then x² + y² is even but not divisible by 9

Consider the two odd positive numbers x and y to be 3k + 1 and 3p + 1.

We can write it as

x² + y² = (3k + 1)² + (3p + 1)²

Using the algebraic identity (a + b)² = a² + b² + 2ab

= 3k² + 3k + 1 + 3p² + 3p + 1

On rearranging

= 3k² + 3p² + 3k + 3p + 2

Taking out 3 as common

= 3(k² + p² + k + p) + 2

The sum of squares is even if the number is not divisible by 9

Therefore, if x and y are odd positive integers, then x² + y² is even but not divisible by nine

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 1

NCERT Exemplar Class 10 Maths Exercise 1.3 Problem 7

Prove that if x and y are both odd positive integers, then x² + y² is even but not divisible by 4

Summary:

If x and y are odd positive integers, then x² + y² is even but not divisible by four.Hence Proved

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