The number of co-primers of 200 lying between 1 to 100 is :

Options

A. 10
B. 40
C. 50
D. None of these

Answer

To find the number of co-primes of 200 lying between 1 and 100, we can follow these steps:

### Step 1: Identify the prime factorization of 200
The first step is to find the prime factorization of 200.
- 200 can be expressed as:
\[ 200 = 2^3 \times 5^2 \]

### Step 2: Identify the range
We need to find the numbers between 1 and 100 that are co-prime to 200. A number is co-prime to another if they share no common factors other than 1.

### Step 3: Identify the multiples of the prime factors
Next, we need to find the multiples of the prime factors of 200 (which are 2 and 5) between 1 and 100.

#### Finding multiples of 2:
- The multiples of 2 between 1 and 100 are:
\[ 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100 \]
- There are 50 multiples of 2.

#### Finding multiples of 5:
- The multiples of 5 between 1 and 100 are:
\[ 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100 \]
- There are 20 multiples of 5.

### Step 4: Apply the principle of inclusion-exclusion
Since some numbers are counted in both multiples of 2 and multiples of 5, we need to subtract the common multiples (which are multiples of 10).

#### Finding multiples of 10:
- The multiples of 10 between 1 and 100 are:
\[ 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 \]
- There are 10 multiples of 10.

### Step 5: Calculate the total number of multiples of 2 or 5
Using the principle of inclusion-exclusion:
\[ \text{Total multiples of 2 or 5} = (\text{Multiples of 2}) + (\text{Multiples of 5}) - (\text{Multiples of 10}) \]
\[ = 50 + 20 - 10 = 60 \]

### Step 6: Calculate the total numbers between 1 and 100
There are a total of 100 numbers between 1 and 100.

### Step 7: Calculate the co-primes
To find the count of numbers that are co-prime to 200:
\[ \text{Co-primes} = \text{Total numbers} - \text{Total multiples of 2 or 5} \]
\[ = 100 - 60 = 40 \]

### Final Answer
The number of co-primes of 200 lying between 1 and 100 is **40**.

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