Question

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

Answer 1

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Answer 2

Let a be the +ve integer in the form of 3p , 3p+1 , 3p+2
By Euclid Division Algorithm
a = bq + r 
The possible remainder are 
r = 0, 1, 2
r=0
a=3q+0
a=3q
C.O.B.S                    cubing on both sides
a cube = [3q ]cube
a cube = [27q ] cube
a cube = 9 [3q cube]
a cube = 9  { where 3q cube = m }
a cube = 9m
 You can substitute  
r = 1 or r = 2 and get the answer 
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