The square of any positive odd integer for some integer m is of the form

Foundation

Mathematics Foundation

Properties of Real Number

Linear Equations in One Variable

Playing with Numbers

Question

The square of any positive odd integer for some integer m is of the form

7m + 1

7m + 2

8m +1

8m + 3

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We know that any positive odd integer n is of the form 4q + 1 or 4q + 3 where q is some integer.
When, n = 4q + 1
n² = (4q + 1)²
= 16q² + 8q + 1
= 8q(2q + 1) + 1
= 8m + 1Where m = q(2q + 1)
=> n² is in the form 8m + 1

When

n = 4 q + 3

n^{2} = (4 q + 3)^{2}

= 16 q^{2} + 24 q + 9

= 16 q^{2} + 24 q + 8 + 1

= 8 (2 q^{2} + 3 q + 1) + 1

= 8 m + 1

Where

m = (2 q^{2} + 3 q + 1)

=&gt;n^2

is in the form

8 m + 1

Hence,

n^{2}

is in the form

8 m + 1

n

is an odd positive integer.