Question

Let \(a_n = 111111\ldots1\), where \(1\) occurs \(n\) number of times. Then, i. \(a_{741}\) is not a prime.
ii. \(a_{534}\) is not a prime.
iii. \(a_{123}\) is not a prime.
iv. \(a_{77}\) is not a prime.

• A. (i) is correct.
• B. (i) and (ii) are correct.
• C. (ii) and (iii) are correct.
• D. All of them are correct.
• E. None of them is correct.

Hint:

For repunits \(a_n = \frac{10^n - 1}{9}\), if \(n\) is composite, then \(a_n\) cannot be prime. Only for prime \(n\), \(a_n\) has a chance of being prime (though not always). Check divisibility rules using factors of \(n\).

Answer 1

The Correct Option is D

Step 1: Understanding \(a_n\).
The number \(a_n\) is a repunit (a number consisting entirely of \(n\) ones). \[ a_n = 111111\ldots1 = \frac{10^n - 1}{9} \] Thus, the properties of \(a_n\) depend on divisibility of \((10^n - 1)\).

Step 2: Analyze case (i): \(a_{741}\).
- \(741\) ones means \(a_{741}\). - Rule: if \(n\) is divisible by \(3\), then \(a_n\) is divisible by \(a_3 = 111\). - Since \(741\) is divisible by \(3\), \(a_{741}\) is divisible by \(3\). \[ \Rightarrow a_{741} \text{ is not prime.} \]

Step 3: Analyze case (ii): \(a_{534}\).
- \(534\) ones means \(a_{534}\). - Again, \(534\) is divisible by \(3\). - Hence, \(a_{534}\) is divisible by \(a_3 = 111\). \[ \Rightarrow a_{534} \text{ is not prime.} \]

Step 4: Analyze case (iii): \(a_{123}\).
- \(123\) ones means \(a_{123}\). - Since \(123\) is divisible by \(3\), \(a_{123}\) is divisible by \(a_3\). \[ \Rightarrow a_{123} \text{ is not prime.} \]

Step 5: Analyze case (iv): \(a_{77}\).
- \(77\) ones means \(a_{77}\). - Here, \(77\) is not divisible by \(3\). But another property: If \(n\) is composite, \(a_n\) is not prime. - Since \(77 = 7 \times 11\), we can write: \[ a_{77} = \frac{10^{77}-1}{9} \] Now, \((10^{11} - 1)\) divides \((10^{77} - 1)\) because \(11 \mid 77\). Thus, \(a_{77}\) is divisible by \(a_{11}\). \[ \Rightarrow a_{77} \text{ is not prime.} \]

Step 6: Conclusion.
All four cases show that each given repunit is not prime. \[ \boxed{\text{All statements (i), (ii), (iii), (iv) are correct.}} \]