If `a in N` and `1 le a le 20`, then find the number of prime numbers in the form of `5a+1`.

Options

A. 4
B. 5
C. 6
D. 7

Answer

To solve the problem of finding the number of prime numbers in the form of \(5a + 1\) where \(a\) is a natural number from 1 to 20, we can follow these steps:

### Step 1: Define the expression
We need to evaluate the expression \(5a + 1\) for all natural numbers \(a\) from 1 to 20.

### Step 2: Calculate \(5a + 1\) for each \(a\)
We will calculate \(5a + 1\) for each integer value of \(a\) from 1 to 20.

- For \(a = 1\): \(5(1) + 1 = 6\)
- For \(a = 2\): \(5(2) + 1 = 11\)
- For \(a = 3\): \(5(3) + 1 = 16\)
- For \(a = 4\): \(5(4) + 1 = 21\)
- For \(a = 5\): \(5(5) + 1 = 26\)
- For \(a = 6\): \(5(6) + 1 = 31\)
- For \(a = 7\): \(5(7) + 1 = 36\)
- For \(a = 8\): \(5(8) + 1 = 41\)
- For \(a = 9\): \(5(9) + 1 = 46\)
- For \(a = 10\): \(5(10) + 1 = 51\)
- For \(a = 11\): \(5(11) + 1 = 56\)
- For \(a = 12\): \(5(12) + 1 = 61\)
- For \(a = 13\): \(5(13) + 1 = 66\)
- For \(a = 14\): \(5(14) + 1 = 71\)
- For \(a = 15\): \(5(15) + 1 = 76\)
- For \(a = 16\): \(5(16) + 1 = 81\)
- For \(a = 17\): \(5(17) + 1 = 86\)
- For \(a = 18\): \(5(18) + 1 = 91\)
- For \(a = 19\): \(5(19) + 1 = 96\)
- For \(a = 20\): \(5(20) + 1 = 101\)

### Step 3: List the results
The values of \(5a + 1\) for \(a = 1\) to \(20\) are:
- \(6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, 81, 86, 91, 96, 101\)

### Step 4: Identify the prime numbers
Now, we need to determine which of these numbers are prime:
- **6**: Not prime (divisible by 2 and 3)
- **11**: Prime
- **16**: Not prime (divisible by 2)
- **21**: Not prime (divisible by 3 and 7)
- **26**: Not prime (divisible by 2)
- **31**: Prime
- **36**: Not prime (divisible by 2)
- **41**: Prime
- **46**: Not prime (divisible by 2)
- **51**: Not prime (divisible by 3 and 17)
- **56**: Not prime (divisible by 2)
- **61**: Prime
- **66**: Not prime (divisible by 2)
- **71**: Prime
- **76**: Not prime (divisible by 2)
- **81**: Not prime (divisible by 3)
- **86**: Not prime (divisible by 2)
- **91**: Not prime (divisible by 7)
- **96**: Not prime (divisible by 2)
- **101**: Prime

### Step 5: Count the prime numbers
The prime numbers identified are: \(11, 31, 41, 61, 71, 101\).

Thus, the total count of prime numbers in the form \(5a + 1\) is **6**.

### Final Answer
The number of prime numbers in the form of \(5a + 1\) where \(1 \leq a \leq 20\) is **6**.

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