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If n is any positive integer,then by Euclid's algorithm, show that n^3+2 can be expressed in the for

Asked by kanz_daruler1 26th April 2010, 8:16 PM
Answered by Expert
Answer:

Dear Student,

By euclid's theorem, we know that for any two integers A and B,

A can be expressed in the form,

A = Bq + r

where q is some integer and r in some integer such that 0 <= r <= (B-1).

 

Putting A=n and B=3, we can conclude that any integer n can be expressed in the form

(3q) or (3q+1) or (3q+2)

 

Now,

when n=3q, then

n3+2 = (3q)3+2 = 27q3+2= 9(3k3)+2 = 9m+2 (writing 3k3 as m, as 3k3 is also an integer).

 

when n=3q+1, then

n3+2 = (3q+1)3+2 = 27q3+1+27q2+9q+2= 9(3k3+3q2+q)+3 = 9m+3 (writing (3k3+3q2+q) as m, as it is also an integer).

 

when n=3q+2, then

n3+2 = (3q+2)3+2 = 27q3+8+54q2+36q+2= 9(3k3+6q2+4q+1)+1 = 9m+1 (writing (3k3+6q2+4q+1) as m, as it is also an integer).

 

Hence, for any integer n, n3+2 can be expressed in the form 9m+1 or 9m+2 or 9m+3.

 

Regards Topperlearning.

 

Answered by Expert 28th April 2010, 2:59 PM
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