ay positive iilegchchow that the square of any positive integer is eit

1.	ay positive iilegchchow that the square of any positive integer is either of the form2, 50 + 3,q +s. Use Euclid's division lemma to show that the squ38. Use E3m or 3m 1 for some integer m.f an odd positive integer can be of the form 6q +1 or 6q +3
Answer» 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 a^2= 9q^2 = 3 x ( 3q^2) = 3m (where m = 3q^2) Case II - a = 3q +1 a^2= ( 3q +1 )^2= 9q^2+ 6q +1 = 3 (3q^2+2q ) + 1 = 3m +1 (where m = 3q^2+ 2q ) Case III - a = 3q + 2 a^2= (3q +2 )^2 = 9q^2+ 12q + 4 = 9q^2+12q + 3 + 1 = 3 (3q^2+ 4q + 1 ) + 1 = 3m + 1 where m = 3q^2+ 4q + 1) From all the above cases it is clear that square of any positive integer ( as in this case a^2) is either of the form 3m or 3m +1. From which book have you taken this question? Please tell us so that we can provide you faster answer.

ay positive iilegchchow that the square of any positive integer is either of the form2, 50 + 3,q +s. Use Euclid's division lemma to show that the squ38. Use E3m or 3m 1 for some integer m.f an odd positive integer can be of the form 6q +1 or 6q +3

Answer»

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

a^2= 9q^2

= 3 x ( 3q^2)

= 3m (where m = 3q^2)

Case II - a = 3q +1

a^2= ( 3q +1 )^2= 9q^2+ 6q +1

= 3 (3q^2+2q ) + 1

= 3m +1 (where m = 3q^2+ 2q )

Case III - a = 3q + 2

a^2= (3q +2 )^2

= 9q^2+ 12q + 4

= 9q^2+12q + 3 + 1

= 3 (3q^2+ 4q + 1 ) + 1

= 3m + 1 where m = 3q^2+ 4q + 1)

From all the above cases it is clear that square of any positive integer ( as in this case a^2) is either of the form 3m or 3m +1.

*From which book have you taken this question? Please tell us so that we can provide you faster answer.*