Question

Use Euclid’s division lemma to show that the cube of any positive
integer is of the form 9m, 9m + 1 or 9m + 8.

Solution:


Let a be any positive integer and b = 3
a = 3q + r, where q ≥ 0 and 0 ≤ r < 3
∴a=3qor3q+1or3q+2
Therefore, every number can be represented as these three forms.
There are three cases.

Case 1: When a = 3q,
a3=(3q)3=27q3=9(3q3)=9m,
Where m is an integer such that m=3q in this solution why can't we take 9 m instead they have taken 3q

Answer 1

Let a and b be any positive integer
such that
a=bq+r
b=3, r=0,1,2 0<=r<b
a=3q,3q+1,3q+2
a^3=(3q)3=27q^3=9(3q^3)=9m where 3q^3 is m
a^3=(3q+1)3=
a^3=(3q+2)3=