what is the least positive integer 'n' for which n4+(n+1)4 is composite.
Answers
Answer:
If n is even, n
4
+4
n
is divisible by 4
∴ It is composite number
If n is odd, suppose n=2p+1, where p is a positive integer
Then n
4
+4
n
=n
4
+4.4
2p
=n
4
+4(2
p
)
4
which is of the form n
4
+4b
4
, where b is a positive integer (=2
p
)
n
4
+4b
4
=(n
4
+4b
2
+4b
4
)−4b
2
=(n
2
−2b
2
)
2
−(2b)
2
=(n
2
+2b+2b
2
)(n
2
−2b+2b
2
)
We find that n
4
+4b
4
is a composite number consequently n
4
+4
n
is composite when n is odd.
Hence n
4
+4
n
is composite for all integer values of n> 1.
Given:
An expression.
To Find:
The least positive integer value of n for which the value is composite.
Solution:
The given problem can be solved by using the trial and error method.
1. A composite number is defined as the number which has more than 2 factors, for example, the number 4 has factors 1,2, and 4.
2. For n = 1, the value of the expression is 5. 5 is a prime number so n = 1 is incorrect.
3. For n= 2, the value of the expression is 97. 97 is a prime number. So, n=2 is incorrect.
4. For n=3 the value of the expression is 337. 337 is a prime number. So, n= 3 is incorrect.
5. For n=4 the value of the expression is 881. 881 is a prime number. So, n=4 is incorrect.
6. For n=5, the value of the expression is 1921. 1921 is not a prime number. So, n= 5 is the least positive value of n for which the given expression is a composite number.
Therefore, for n=5 the value of the expression is a composite number.