INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More 

April 15, 2020

Composite number Problem | B.Stat Objective | TOMATO 75

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


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

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.

Subscribe to Cheenta at Youtube


One comment on “Composite number Problem | B.Stat Objective | TOMATO 75”

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
enter