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Composite number Problem | B.Stat Objective | TOMATO 75

Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Logic True-False Reasoning. You may use sequential hints.

Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Sequence and composite number.

Composite number Problem (B.Stat Objective)


Consider the sequence \(a_1\)=101, \(a_2\)=10101,\(a_3\)=1010101 and so on. Then \(a_k\) is a composite number ( that is not a prime number)

  • if and only if \(k \geq 2\) and \(11|(10^{k+1}+1)\)
  • if and only if \(k \geq 2\) and k-2 is divisible by 3
  • if and only if \(k \geq 2\) and \(11|(10^{k+1}-1)\)
  • if and only if \(k \geq 2\)

Key Concepts


Logic

Sequence

Composite number

Check the Answer


But try the problem first…

Answer: if and only if \(k \geq 2\) and k-2 is divisible by 3

Source
Suggested Reading

B.Stat Objective Question 75

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


First hint

for \(a_k\) \(k \geq 2\) may be prime also then not considering this here

Second Hint

for \(a_{8}\) \(10^{9}-1\) and \(10^{9}+1\) not divisible by 11

Final Step

8-2 is divisible by 3 and \(a_{8}\) is composite number then \(a_{k}\) is composite if and only if \(k \geq 2\) and k-2 is divisible by 3.

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