1.

Use Euclid's division lemma to show that the square of any positive integer is either ofthe form 3m or 3m + 1 for some integer m. ​

Answer»

GIVEN,

For any given two integers a and b , there exists a UNIQUE integers between them q and r ,which satisfies

a=bq+r , where ZERO is less than or equal to r

Here b=3

Case 1:

If r=0

a^2=(3q+0)^2

=(3q)^2

=9q^2

=3(3q^2)

= 3m [where m=3q^2]

Case 2:

If r=1

a^2=(3q+1)^2

=(9q^2 + 2×3q×1 + 1)

=(9q^2 + 6q + 1)

=3( 3q^2 + 2q ) + 1

=3m +1 [where m= 3q^2 + 2q]

therefore SQUARE of any positive integer is either of

the form 3m or 3m + 1 for some integer m.


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