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.]

NCERT Class X
Mathematics - Mathematics

Chapter _Real Numbers

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2015-06-10T08:50:26+05:30

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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

a=3q

squaring both sides

a2=9q2=3(3q2)=3m

              where m=3q2

when r=1

a=3q+1

squaring both sides

a2=9q2+1+6q

    =3(3q2+2q)+1

    =3m+1

when r=2

a=3q+2

squaring both sides

a2=9q2+4+12q

     =9q2+3+1+12q

     = 3(3q2+1+4q)+1

     =3m+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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