5. Use Euclid's division lemma to show that the cube of any positive integer is of the form

9m, 9m + 1 or 9m+8.

2 The Fundamental Theorem of Arithmetic

according to euclids division lemma

a=bq+r

Step-by-step explanation:

so, b=3. r=0,1,2

a=3q+r

cubing both sides

a³ = (3q+0) ³

a=3q³

a=9q³ where m = q³

a=9m

r=1

a³ = (3q+1) ³ (a+b) ³=a³+b³+3a²b+3ab²)

a³ = (3q) ³+1³+3 (3q) ² (1) + 3 (3q) (1) ²

=27q³+1+9q²+9q

take common

=9 (3q³+q²+q) + 1. where (3q³+q²+q) = m

=9m+1

r=2

a³ = (3q+2) ³

= (3q) ³+2³+3 (3q) ²+2) + 3 (3q) (2) ²

=27q³+8+54q²+36q

taking common

=9 (3q³+6q²+4q) + 8 where (3q³+6q²+4q) = m

9m + 8