Question

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

Answer 1

Step-by-step explanation:

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 integer m.

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

Thus, we have:

(6m+1)

2

=36m

2

+12m+1

=6(6m

2

+2m)+1

=6q+1,qisaninteger.

(6m+3)

2

=36m

2

+36m+9

=6(6m

2

+6m+1)+3

=6q+3,qisaninteger

(6m+5)=36m

2

+60m+25

=6(6m

2

+10m+4)+1

=6q+1,qisaninteger.

∴ Any positive odd integer is of the form 6q+1 or 6q+3. where q belongs to integers and real.

Answer 2

Answer:

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