Question

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

Answer 1

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

   

Answer 2

Step-by-step explanation:

let ' a' be any positive integer and b = 3.

we know, a = bq + r , 0 <  r< b.

now, a = 3q + r , 0<r < 3.

the possibilities of remainder = 0,1 or 2

Case I - a = 3q

a² = 9q² .

 = 3 x ( 3q²)

 = 3m (where m = 3q²)

Case II - a = 3q +1

a² = ( 3q +1 )²

 =  9q² + 6q +1

 = 3 (3q² +2q ) + 1

 = 3m +1 (where m = 3q² + 2q )

Case III - a = 3q + 2

a² = (3q +2 )²

   = 9q² + 12q + 4

   = 9q² +12q + 3 + 1

 = 3 (3q² + 4q + 1 ) + 1

 = 3m + 1 where m = 3q2 + 4q + 1)

From all the above cases it is clear that square of any positive integer ( as in this case a² ) is either of the form 3m or 3m +1.

Hence, it is solved .

THANKS

#BeBrainly .