For how many positive integers n is it true that the sum of 13/n, 18/n and 29/n is an integer?

Options

A. 6
B. 60
C. Greatest common factor of 13,18 and 29
D. 12

Answer

To solve the problem of finding how many positive integers \( n \) make the sum \( \frac{13}{n} + \frac{18}{n} + \frac{29}{n} \) an integer, we can follow these steps:

### Step 1: Combine the Fractions
First, we combine the fractions since they have the same denominator:
\[ \frac{13}{n} + \frac{18}{n} + \frac{29}{n} = \frac{13 + 18 + 29}{n} \]

### Step 2: Calculate the Numerator
Next, we calculate the sum in the numerator:
\[ 13 + 18 = 31 \]
\[ 31 + 29 = 60 \]
Thus, we have:
\[ \frac{60}{n} \]

### Step 3: Determine When the Fraction is an Integer
We need \( \frac{60}{n} \) to be an integer. This occurs when \( n \) is a divisor of 60. Therefore, we need to find all positive integers \( n \) such that \( n \) divides 60.

### Step 4: Find the Divisors of 60
To find the divisors of 60, we can start by determining its prime factorization:
\[ 60 = 2^2 \times 3^1 \times 5^1 \]
Using the formula for the number of divisors, if \( n = p_1^{k_1} \times p_2^{k_2} \times \ldots \times p_m^{k_m} \), the number of divisors \( d(n) \) is given by:
\[ d(n) = (k_1 + 1)(k_2 + 1) \ldots (k_m + 1) \]
For 60, we have:
- \( k_1 = 2 \) (for \( 2^2 \))
- \( k_2 = 1 \) (for \( 3^1 \))
- \( k_3 = 1 \) (for \( 5^1 \))

Thus, the number of divisors is:
\[ (2 + 1)(1 + 1)(1 + 1) = 3 \times 2 \times 2 = 12 \]

### Step 5: List the Divisors
The positive divisors of 60 are:
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

### Conclusion
Therefore, there are a total of 12 positive integers \( n \) such that \( \frac{60}{n} \) is an integer.

The final answer is:
\[ \boxed{12} \]