Find how many odd factors are there in the number 1240.

Options

A. 2
B. 4
C. 5
D. 16

Answer

To find how many odd factors are there in the number 1240, we will follow these steps:

### Step 1: Prime Factorization of 1240
First, we need to perform the prime factorization of 1240.

1. Divide by 2 (the smallest prime number):
- 1240 ÷ 2 = 620
- 620 ÷ 2 = 310
- 310 ÷ 2 = 155
- 155 is not divisible by 2, so we move to the next prime number, which is 5.
- 155 ÷ 5 = 31
- 31 is a prime number.

So, the prime factorization of 1240 is:
\[ 1240 = 2^3 \times 5^1 \times 31^1 \]

### Step 2: Identify Odd Factors
To find the odd factors, we need to ignore the even prime factor (which is 2). Thus, we only consider the odd prime factors, which are 5 and 31.

### Step 3: Calculate the Number of Odd Factors
The formula to find the number of factors from the prime factorization is:
If \( n = p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k} \), then the number of factors is given by:
\[ (e_1 + 1)(e_2 + 1)...(e_k + 1) \]

For the odd part of 1240, we have:
- \( 5^1 \) contributes \( 1 + 1 = 2 \) factors (which are 1 and 5).
- \( 31^1 \) contributes \( 1 + 1 = 2 \) factors (which are 1 and 31).

### Step 4: Total Odd Factors
Now, we multiply the number of factors from the odd prime factors:
\[ \text{Total odd factors} = (1 + 1)(1 + 1) = 2 \times 2 = 4 \]

### Conclusion
The total number of odd factors of 1240 is 4.

### Final Answer
Thus, the answer is 4.

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