In a set of first 180 natural numbers find the number of prime numbers.

Options

A. A)48
B. B)24
C. C)41
D. D)38

Answer

To find the number of prime numbers in the first 180 natural numbers, we can use the information about the number of prime numbers in ranges of natural numbers. Here’s a step-by-step solution:

### Step 1: Understand the range of natural numbers
The first 180 natural numbers are the numbers from 1 to 180.

### Step 2: Recall the number of prime numbers in related ranges
From the information provided:
- There are 15 prime numbers between 1 and 50.
- There are 25 prime numbers between 1 and 100.
- There are 46 prime numbers between 1 and 200.

### Step 3: Identify the prime numbers between 180 and 200
We need to find how many prime numbers exist between 180 and 200. The prime numbers in this range are:
- 181
- 191
- 193
- 197
- 199

Counting these, we find there are 5 prime numbers.

### Step 4: Calculate the number of prime numbers from 1 to 180
Since we know there are 46 prime numbers from 1 to 200, we can subtract the number of prime numbers from 181 to 200 (which is 5) from this total.

\[ \text{Number of primes from 1 to 180} = \text{Total primes from 1 to 200} - \text{Primes from 181 to 200} \]
\[ = 46 - 5 = 41 \]

### Final Answer
Thus, the number of prime numbers in the first 180 natural numbers is **41**.

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