The number of divisors of 187, 637 and 1001 are in:

Options

A. A.P.
B. G.P.
C. H.P.
D. none

Answer

To solve the problem of determining the relationship between the number of divisors of the numbers 187, 637, and 1001, we will follow these steps:

### Step 1: Find the prime factorization of 187
187 can be factored as:
\[ 187 = 11 \times 17 \]
Both 11 and 17 are prime numbers.

### Step 2: Calculate the number of divisors of 187
Using the formula for the number of divisors:
\[ \text{Number of divisors} = (e_1 + 1)(e_2 + 1)...(e_n + 1) \]
where \( e_i \) are the powers of the prime factors.

For 187:
- The prime factorization is \( 11^1 \times 17^1 \).
- Therefore, the number of divisors is:
\[ (1 + 1)(1 + 1) = 2 \times 2 = 4 \]

### Step 3: Find the prime factorization of 637
637 can be factored as:
\[ 637 = 7 \times 91 = 7 \times 7 \times 13 = 7^2 \times 13 \]

### Step 4: Calculate the number of divisors of 637
For 637:
- The prime factorization is \( 7^2 \times 13^1 \).
- Therefore, the number of divisors is:
\[ (2 + 1)(1 + 1) = 3 \times 2 = 6 \]

### Step 5: Find the prime factorization of 1001
1001 can be factored as:
\[ 1001 = 7 \times 11 \times 13 \]

### Step 6: Calculate the number of divisors of 1001
For 1001:
- The prime factorization is \( 7^1 \times 11^1 \times 13^1 \).
- Therefore, the number of divisors is:
\[ (1 + 1)(1 + 1)(1 + 1) = 2 \times 2 \times 2 = 8 \]

### Step 7: List the number of divisors
Now we have:
- Number of divisors of 187 = 4
- Number of divisors of 637 = 6
- Number of divisors of 1001 = 8

### Step 8: Determine the relationship between the numbers
The numbers of divisors are 4, 6, and 8. We can check if they are in an arithmetic progression (AP):
- The common difference between 4 and 6 is 2.
- The common difference between 6 and 8 is also 2.

Since the common difference is the same, the numbers 4, 6, and 8 are in AP.

### Conclusion
The number of divisors of 187, 637, and 1001 are in Arithmetic Progression (AP).

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