Question

Choose the prime number from 175, 161, 129, 101

• A. 161
• B. 101
• C. 175
• D. 129

Hint:

Familiarizing yourself with prime numbers up to \(200\) can save crucial time.
Common tricky composites to watch out for in exams are \(119\) (\(7 \times 17\)), \(133\) (\(7 \times 19\)), and \(161\) (\(7 \times 23\)).
Always test the prime divisibility rule up to \(\sqrt{N}\) to be absolutely sure.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The topic of this question is the Number System, specifically Prime and Composite Numbers.
A prime number is a natural number greater than \(1\) that has no positive divisors other than \(1\) and itself.
A composite number has other factors besides \(1\) and itself.
We need to evaluate the given options to find which one is prime.

Step 2: Key Formula or Approach:
To check if a number \(N\) is prime:
Find the approximate square root of \(N\), say \(k = \sqrt{N}\).
Test divisibility of \(N\) by all prime numbers less than or equal to \(k\) (i.e., \(2, 3, 5, 7, \dots\)).
If \(N\) is not divisible by any of these prime numbers, then \(N\) is prime.

Step 3: Detailed Explanation:

• Let us analyze each of the given options step-by-step:

• Analyzing 175:
- The units digit of \(175\) is \(5\).
- Any number ending in \(5\) is divisible by \(5\).
- Specifically, \(175 = 5 \times 35\).
- Since it has divisors other than \(1\) and \(175\), it is a composite number.

• Analyzing 161:
- Let us check divisibility by small prime numbers.
- It is not even, so it is not divisible by \(2\).
- The sum of the digits is \(1 + 6 + 1 = 8\), which is not divisible by \(3\).
- It does not end in \(0\) or \(5\), so it is not divisible by \(5\).
- Let us test divisibility by \(7\):
\[ \frac{161}{7} = 23 \]
- Since \(161 = 7 \times 23\), it is a composite number.

• Analyzing 129:
- The sum of the digits is \(1 + 2 + 9 = 12\).
- Since the sum of the digits is divisible by \(3\), the number \(129\) is divisible by \(3\).
- Specifically, \(129 = 3 \times 43\).
- Thus, \(129\) is a composite number.

• Analyzing 101:
- Let us test prime numbers up to \(\sqrt{101} \approx 10\). The primes to check are \(2, 3, 5\), and \(7\).
- \(101\) is not divisible by \(2\) (it is odd).
- \(101\) is not divisible by \(3\) (sum of digits is \(2\)).
- \(101\) is not divisible by \(5\) (ends in \(1\)).
- \(101\) is not divisible by \(7\) (\(101 = 7 \times 14 + 3\)).
- Since it is not divisible by any prime up to its square root, \(101\) is a prime number.


Step 4: Final Answer:
The prime number in the given options is \(101\), which corresponds to Option (B).