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I.S.I. and C.M.I. Entrance

Prime Number for ISI BStat | TOMATO Objective 70

Try this beautiful problem from Prime number from TOMATO useful for ISI BStat Entrance. You may use sequential hints to solve the problem.

Try this beautiful problem from Integer based on Prime number useful for ISI BStat Entrance.

Prime number | ISI BStat Entrance | Problem no. 70


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

Key Concepts


Number theory

Algebra

Prime

Check the Answer


But try the problem first…

Answer: One

Source
Suggested Reading

TOMATO, Problem 70

Challenges and Thrills in Pre College Mathematics

Try with Hints


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\((3,5,7)\)

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