CBSE Class 10 Answered

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.