A positive integer is in the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1 i.e., 3m or 3m on+ 2 for some integer m? Justify your answer.
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Answer:
By Euclid's division algorithm , a=bq+r where a,b,q,r are non-negative integers and 0≤r<b.
On putting b=3 and r=1 we get
a=3q+1
Squaring both sides
⇒a
2
=(3q+1)
2
⇒a
2
=(3q)
2
+(1)
2
+2(3q)
⇒a
2
=3(3q
2
+2q)+1
⇒a
2
=3m+1 , where $$ m = 4q^2 + 2q $ is any integer.
Hence, the square of a positive integer iof the form 3q+1 can not be written in any form other than 3m+1
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