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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 ......

- Zero
- One
- Infinite
- More than one,but finite

Number theory

Algebra

Prime

But try the problem first...

Answer: One

Source

Suggested Reading

TOMATO, Problem 70

Challenges and Thrills in Pre College Mathematics

First hint

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 ..........

Second Hint

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........

Final Step

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\((3,5,7)\)

- https://www.cheenta.com/problem-on-semicircle-amc-8-2013-problem-20/
- https://www.youtube.com/watch?v=wStgy7cRAfQ

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