Question

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.
3 ​

Answer 1

Let 'a' be a positive integer and b = 9

By Euclid's division lemma a = bq + r where 0 is less than or equal to r and r is less than b

So, r = 0,1,2,3,4,5,6,7,8

if r = 0 then a = 9q + 0

a^3=(9q)^3

=729q^3

=9(81^3)

a^3=9m [m is some integer]

if r =1 then a = 9q+1

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

=729 q^3+1+243q^2+27q

=9(81q^3+27q^2+3q)+1

a^3=9m+1 [m is some integer]

if r = 2 then a = 9q+2

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

=729q^3+8+486q^2+108q

=9(81q^3+54q^2+12q)+8

a^3=9m+8 [ m is some integer]

if we continue this process ,then we have 9m,9m+1 or 9m+8

So, the cube of any positive integer is of the form 9m,9m+1 or 9m+8

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