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Use Euclid’s division lemma to show that the square of any positive integer is either of

the form 3m or 3m + 1 for some integer m.
[Hint : Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square
each of these and show that they can be rewritten in the form 3m or 3m + 1.]

Mathematics - Mathematics

Chapter _Real Numbers



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let a be any +ve integer.and it is of the form 3q,3q+1,3q+2

by euclids division algorithm;

a = bq +r, here b=3

so r=0,1,2

when r=0


squaring both sides


              where m=3q2

when r=1


squaring both sides




when r=2


squaring both sides



     = 3(3q2+1+4q)+1


where m=3q2+1+4q

this shows that square of any +ve integer is either of the form 3m or 3m+1 for some integer m


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