Question

. Using Euclid's division theorem, show that the cube of a positive integer. 9m is of the form 9m + 1 or 9m + 8. ​

Answer 1

let a and b be any number

now ,

according to Euclid division algorithm

a=bq+r

we have to take b =3 (to have composite) r=0

a = (3q +o )3

a= (27q)= 9(3q) = 9m (3q =m)

on taking r=1

a= (3q +1 )3

a=a3+b3+3a2b +3ab2

a= (3q) 3 +(1)3+ 3.3qsquare.1+3.3q.1 square

a= 27q3 + 1 + 27q square + 9q square

a= 9 q( 3q2 + 3q +1q) +1 (m = q(3q2+3q+1q) )

a= 9m +1..

a=(3q+2)3

a = 3q3+ 2cube + 3.3q2. 2 +3.3q.2square

a=27q3 +8 + 54q2+ 36q2

a=9q(3q2+ 5q2+3q2) +8

a= 9m +8 ( q(3q2+5q2+3q2) = m )

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