Question

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the Euclid's divon lemma show that the cube of any intergers of any integers is of the 9m,9m+1or9m + 8 ​

Answer 1

Step-by-step explanation:

let 'a' be a positive integer b=3

apply division algorithm for a and b=3

a=3q+r , q>0 , 0 is less than or equal to r <3

the possible remainders are 0 , 1 , 2

a=3q+0

a=3q

a=3q+1

a=3q+2

case 1:- a=3q

cubing on both sides

a³=(3q)³

a³=27q³

a³= 9(3q³)

let 3q³= m

a³=9m

case 2:- a=3q+1

cubing on both sides

a³=(3q+1)³

a³= (3q)³+3(3q)²(1)+3(3q)(1)²+1³

a³=27q³+3(9q²)+9q+1

a³= 27q³+27q²+9q+1

a³= 9(3q³+3q²+q)+1

let 3q³+3q²+q= m

a³= 9m+1

case 3:- a=3q+2

cubing on both sides

a³=(3q+2)³

a³= (3q)³+3(3q)²(2)+3(3q)(2)²+2³

a³= 27q³+3(9q²)(2)+3(3q)(4)+8

a³= 27q³+54q²+36q+8

a³= 9(3q³+6q²+4q)+8

let 3q³+6q²+4q= m

a³= 9m+8

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