Among the first 100 even natural numbers haw many numbers have even number of factors?

Options

A. 37
B. 50
C. 73
D. 93

Answer

To solve the problem of how many even natural numbers among the first 100 even natural numbers have an even number of factors, we can follow these steps:

### Step 1: Identify the first 100 even natural numbers
The first 100 even natural numbers are:
2, 4, 6, 8, ..., 200.
This sequence can be represented as:
\[ 2n \quad \text{for} \quad n = 1, 2, 3, \ldots, 100 \]

### Step 2: Understand the concept of factors
A number has an even number of factors if it is not a perfect square. This is because factors generally come in pairs, and only perfect squares have an unpaired factor (the square root).

### Step 3: Identify perfect squares among the first 100 even natural numbers
The perfect squares among the first 100 even natural numbers are:
- The even perfect squares up to 200 are: \( 4 (2^2), 16 (4^2), 36 (6^2), 64 (8^2), 100 (10^2), 144 (12^2), 196 (14^2) \).
- These correspond to \( 2^2, 4^2, 6^2, 8^2, 10^2, 12^2, 14^2 \).

### Step 4: Count the perfect squares
The perfect squares identified are:
- 4, 16, 36, 64, 100, 144, 196.
This gives us a total of 7 perfect squares.

### Step 5: Calculate the total numbers with even factors
Since there are 100 even natural numbers in total and 7 of them are perfect squares (which have an odd number of factors), the numbers with an even number of factors are:
\[ 100 - 7 = 93 \]

### Final Answer
Thus, the number of even natural numbers among the first 100 even natural numbers that have an even number of factors is **93**.

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