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 Apr, 2010, 08:16: PM
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)
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.
Answered by | 28th Apr, 2010, 02:59: PM
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