Question

use division algorithm to show that the cube of any positive integer is of the form 9m,9m+1 or 9m+8​

Answer 1

Step-by-step explanation:

Let "y" be any positive Integers , then "y" is of the form 3q, (3q +1) or (3q + 2),where q is some integer.

consider

y = 3q

y3 =(3q)3

= 27 q3

= 9 (3qcube)

y3 = 9m (where m = 3q3)

y = ( 3q + 1)

y3 = (3q + 1 )cube

= 27q3 + 27q2+9q+1

= (3q3 + 3q2 +q )+ 1

y3 = 9m ( where m = 3q3 + 3q2 +q)

y = (3q + 2)

y3 = (3q +2)3

= 27q3 + 54 q2 + 36q + 8

= 9 ( 3q3 + 6 q2 + 4q ) + 3

y3 = 9m + 8 ( where m = 3q3 + 6q2 +4q )

y3 = 9m or (9m +1) or (9m + 3)

therefore , the cube of any positive integer is either of the form 9m or (9m +1) or (9m + 8).

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