n3 + 1 can be expressed in the form 9 m 9 m + 1 or 9 m + 2.
Here, m
m is some integer.
Answers
Answer:
Let n be any positive integer and b=3
According to Euclid's division algorithm there exist integers q and r such that
n=3×q+r where 0≤r<3
So value of r can be 0,1 and 2
We can write
n=3×q
, n=3×q+1 and n=3×q+2
Now take the each case one by one
Case 1: n=3×q
Take cube on both sides, we have
n
3
=27×q
3
= 9(3×q
3
)=9m where m=3q
3
∴n
3
+1=9m+1
Similarly for 2nd case n=3×q+1
Taking cube on both sides, we have
n
3
=27q
3
+27q
2
+9q+1 or n
3
=9(3q
3
+3q
2
+q)+1
Which is equal to 9m+1 where m=(3q
3
+3q
2
+q)
Thus, n
3
+1=9m+2
For final case n=3×q+2
Taking cube on both sides, we have
n
3
=27q
3
+54q
2
+36q+8
i.e. n
3
=9(3q
3
+6q
2
+4q)+8
which is equal to 9m+8 where m=(3q
3
+6q
2
+4q)
Hence, n
3
+1=9m+8+1=9m+9=9(m+1)=9n where n=m+1 and n is an integer