show that the squre of any positive integer is of the form 3m or 3m+1 for some integer m.

Asked by FEHMEEDAH NAZNEEN | 30th Aug, 2010, 08:49: PM

Expert Answer:

we use Euclid's lemma here.
so taking b=3,
we get,
a=3q+r where 0 less than or equal to r less than 3 and q is greater than or equal to zero.
i.e. r=0,1,2
so
a=3q
or
a=3q+1
or
a=3q+2
so squaring
a=9q2=3(3q2)
or
a=(3q+1)2=9q2+6q+1=3(3q2+2q)+1
or
a=(3q+2)2=9q2+12q+4=3(3q2+4q+1)+1
so in either case a takes the form as mentioned in the question as q is a on negative  integer.
 

Answered by  | 31st Aug, 2010, 10:41: AM

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