Use euclid division lemma to show that the cube of any positive integer is of the form 9m, 9m+1,9m+8

Use euclid division lemma to show that the cube of any positive intege

1.	Use euclid division lemma to show that the cube of any positive integer is of the form 9m, 9m+1,9m+8
Answer» Let a = 3q + r\xa0{tex}: 0 \\leq r < 3{/tex}\xa0{tex}\\therefore \\quad a = 3 q ; \\text { then } a ^ { 3 } = 27 q ^ { 3 } = 9 m ; \\text { where } m = 3 q ^ { 3 }{/tex}{tex}\\text { when } a = 3 q + 1 ; \\text { then } a = 27 q ^ { 2 } + 27 q ^ { 2 } + 9 q + 1{/tex}{tex}= 9 \\left( 3 q ^ { 3 } + 3 q ^ { 2 } + q \\right) + 1{/tex}{tex}= 9 m + 8 \\quad \\left( \\text { where } m = 3 q ^ { 3 } + 3 q ^ { 2 } + q \\right){/tex}{tex}\\text { when } a = 3 q + 2 ; \\text { then } a ^ { 3 } = ( 3 q + 2 ) ^ { 2 }{/tex}{tex}= 27 q ^ { 3 } + 54 q ^ { 2 } + 36 q + 8{/tex}{tex}= 9 m + 8 \\quad \\left( \\text { where } m = 3 q ^ { 3 } + 6 q ^ { 2 } + 4 q \\right){/tex}{tex}= 9 m + 8 \\quad \\left( \\text { where } m = 3 q ^ { 3 } + 6 q ^ { 2 } + 4 q \\right){/tex}Hence, cubes of any positive integer is\xa0either of the from 9m, (9m + 1) or (9m + 8).