Try this beautiful problem from Integer based on Prime number useful for ISI BStat Entrance.
The number of integers \(n>1\), such that n, n+2, n+4 are all prime numbers is ......
Number theory
Algebra
Prime
Answer: One
TOMATO, Problem 70
Challenges and Thrills in Pre College Mathematics
taking n=3, 5, 7, 11, 13, 17....prime numbers we will get
Case of n=3
n= 3
n+2=5
n+4=7
Case of n=5
then \(n\)=5
n+2=7
n+4=9 which is not prime....
Case of n=7,
then n=7
n+2=9 which is not prime ...
n+4=11
Can you now finish the problem ..........
We observe that when n=3 then n,n+2,n+4 gives the prime numbers.....other cases all are not prime.Therefore any no can be expressed in anyone of the form 3k, 3k+1 and 3k+2.
can you finish the problem........
If n is divisible by 3 , we are done. If the remainder after the division by 3 is 1, the number n+2 is divisible by 3. If the remainder is 2, the number n+4 is divisible by 3
The three numbers must be primes! The only case n=3 and gives\((3,5,7)\)
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