Column A: The number of prime numbers between 70 and 76
Column B: The number of prime numbers between 30 and 36
Column B: The number of prime numbers between 30 and 36
Show Hint
- Quantity A is greater
- Quantity B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given
The Correct Option is A
Solution and Explanation
This question requires identifying prime numbers within two specific ranges. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
Step 2: Detailed Explanation:
Analyzing Column A:
We need to find the prime numbers between 70 and 76. The integers in this range are 71, 72, 73, 74, 75.
- 71: To check if 71 is prime, we can test for divisibility by primes up to \(\sqrt{71}\) (which is approx 8.4). Primes to test are 2, 3, 5, 7.
- Not divisible by 2 (it's odd).
- Not divisible by 3 (sum of digits 7+1=8, not div by 3).
- Not divisible by 5 (doesn't end in 0 or 5).
- Not divisible by 7 (\(7 \times 10 = 70\)).
So, 71 is prime.
- 72: Divisible by 2 (it's even). Not prime.
- 73: To check if 73 is prime, we test primes 2, 3, 5, 7.
- Not divisible by 2, 3, or 5.
- Not divisible by 7 (\(7 \times 10 = 70\)).
So, 73 is prime.
- 74: Divisible by 2. Not prime.
- 75: Divisible by 5. Not prime.
The number of prime numbers in this range is 2.
Analyzing Column B:
We need to find the prime numbers between 30 and 36. The integers in this range are 31, 32, 33, 34, 35.
- 31: To check if 31 is prime, we test primes up to \(\sqrt{31}\) (approx 5.5). Primes to test are 2, 3, 5. - Not divisible by 2, 3, or 5.
So, 31 is prime.
- 32: Divisible by 2. Not prime.
- 33: Divisible by 3 (\(3 \times 11 = 33\)). Not prime.
- 34: Divisible by 2. Not prime.
- 35: Divisible by 5. Not prime.
The number of prime numbers in this range is 1.
Step 3: Comparing the Quantities:
Column A: 2
Column B: 1
Since \(2 \textgreater 1\), the quantity in Column A is greater.
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