Q. Show that any positive integer is of the form 3q or 3q+1 or 3q+2 for some integer q.

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Solution

Let x be the integer
x=3q
x²=9q²
x²=3(3q²)
x²=3q [let 3q² be q]

x=3q+1
x²=(3q+1)²
x²=9q²+6q+1
x²=3(3q²+2q)+1
x²=(3q+1) [let 3q²+2q be q]

x=3q+2
x²=(3q+2)²
x²=9q²+12q+4
x²=3(3q²+4q+1)+1
x²=(3q+1) [ let 3q²+4q+1 be q]

It proves q² is in the form of 3q and (3q+1) but not in the form of (3q+2)

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Say true or false:

A positive integer is of the form 3q + 1, q being a natural number, then you write its square in any form other than 3m + 1, i.e.,3m or 3m + 2 for some integer m.

Q. A positive integer is of the form

3 q + 1, q

being a anatural number. Can you write its square in any form other than

3 m + 1, 3 m

3 m + 2

for some integer

m

? Justify your answer.

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