Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q

Solution:

We know that,

Any positive integer is of the form 6m, 6m + 1, 6m + 2, 6m + 3, 6m + 4 or 6m + 5, for some integers m.

Any odd positive integer can be of the form 6m + 1, 6m + 3, or 6m + 5.

Now,

(6m + 1)² = 36m² + 12 m + 1

= 6(6m² + 2 m) + 1

= 6q + 1, q is an integer.

(6m + 3)² = 36m² + 36 m + 9

= 6(6m² + 6 m + 1) + 3

= 6q + 3, q is an integer.

(6m + 5)² = 36m² + 60 m + 25

= 6(6m² + 10 m + 4) + 1

= 6q + 1, q is an integer.

Therefore, the square of an odd positive integer can be of the form 6q + 1 or 6q + 3

✦ Try This: Show that the square of an odd positive integer can be of the form 6p + 1 or 6p + 3 for some integer p

We know that

Any positive integer can be of the form 6m, 6m + 1, 6m + 2, 6m + 3, 6m + 4 or 6m + 5, for some integers m.

An odd positive integer can be of the form 6m + 1, 6m + 3, or 6m + 5

Now:

(6m +1)² = 36m² + 12m + 1

= 6(6m² + 2m) + 1

= 6q + 1, p is an integer

(6m + 3)² = 36m² + 36m + 9

= 6(6m² + 6m + 1) + 3

= 6q + 3, p is an integer

(6m + 5)² = 36m² + 60m + 25

= 6(6m² + 10m + 4) + 1

= 6q + 1, p is an integer

Therefore, the square of an odd positive integer can be of the form 6p + 1 or 6p + 3

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

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NCERT Exemplar Class 10 Maths Exercise 1.4 Solved Problem 1

Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q

Summary:

The square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q. Hence proved

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☛ Related Questions:

• Show that the cube of a positive integer of the form 6q + r, q is an integer and r = 0, 1, 2, 3, 4, . . . .
• Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive int . . . .
• Prove that one of any three consecutive positive integers must be divisible by 3