Question

Show that a cube of a positive integer is of the form 6q+r where q is a integer and r=0,1,2,3,4,5 is also of the form 6m+r.

Answer 1

Answer:

a=6q+r it's cube is

a^3=216q^3+r^3+18qr(6q+r)

a^3=216q^3+r^3+108q^2+18qr^2

Case 1 = when r=0 then a^3=216q^3(6m here m =36q^3)

Case 2=when r =1 then a^3 = 216q^3 +108q^2+18q +1(6m+1 here m = 36q^3+18q^2+3q)

Hence the cube of any positive integer is either to the form 6m , 6m+1 or 6m+2 --------

Other case you can do yourself.

I hope it helps you.

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