Question

Using division algorithm to show that the cube of positive integer is in the form of 9m,9m+1,9m+8.

Answer 1

by Euclid's division algorithm we have a=bq+r where 0›=r›b ---------(1)
put b=9 in eqn 1
a=bq+r
a=9q+r
r{0,1,2,.................8}
if r= 0
a=9q+r
a=9q+0
a=9q
if r= 1
a=9q+r
=9q+1
if r = 2
a=9q+r
=9q+ 2
putting cube on these
a=9q
a3=9q3
a3=729q3
a3=9(81q3).
a3=9m. (where m=81q3)

a=9q+1
a3=(9q+1)3
= 729q3+1
a3=9(81q3)+1
a3=9m+1
a=9q+2
a3=(9q+2)3
=729q3+8
=9(81q3)+8
a3=9m+8

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