The prime factorization of a number 'n' is given as `n=2^x xx 3^y xx 5^z , y+z=5 and y^-1+z^-1 =5/6` . Find out the odd divisors of n including 1

Answer

To find the odd divisors of the number \( n \) given its prime factorization \( n = 2^x \times 3^y \times 5^z \), we start with the conditions provided:

1. \( y + z = 5 \) (Equation 1)
2. \( \frac{1}{y} + \frac{1}{z} = \frac{5}{6} \) (Equation 2)

### Step 1: Solve for \( y \) and \( z \)

From Equation 2, we can rewrite it as:

\[ \frac{z + y}{yz} = \frac{5}{6} \]

Substituting \( y + z = 5 \) into the equation gives:

\[ \frac{5}{yz} = \frac{5}{6} \]

Cross-multiplying yields:

\[ 5 \cdot 6 = 5 \cdot yz \implies yz = 6 \quad \text{(Equation 3)} \]

### Step 2: Solve the system of equations

Now we have two equations:

1. \( y + z = 5 \) (Equation 1)
2. \( yz = 6 \) (Equation 3)

We can express \( z \) in terms of \( y \) from Equation 1:

\[ z = 5 - y \]

Substituting this into Equation 3:

\[ y(5 - y) = 6 \]

Expanding this gives:

\[ 5y - y^2 = 6 \implies y^2 - 5y + 6 = 0 \]

### Step 3: Factor the quadratic equation

Factoring the quadratic:

\[ (y - 2)(y - 3) = 0 \]

Thus, the solutions are:

\[ y = 2 \quad \text{or} \quad y = 3 \]

### Step 4: Find corresponding values of \( z \)

Using \( y + z = 5 \):

- If \( y = 2 \), then \( z = 5 - 2 = 3 \).
- If \( y = 3 \), then \( z = 5 - 3 = 2 \).

Thus, we have two pairs of values: \( (y, z) = (2, 3) \) and \( (3, 2) \).

### Step 5: Determine the odd divisors of \( n \)

The odd divisors of \( n \) are determined by the factors \( 3^y \) and \( 5^z \). The number of odd divisors can be calculated using the formula:

\[ \text{Number of odd divisors} = (y + 1)(z + 1) \]

Calculating for both pairs:

1. For \( (y, z) = (2, 3) \):

\[ \text{Number of odd divisors} = (2 + 1)(3 + 1) = 3 \times 4 = 12 \]

2. For \( (y, z) = (3, 2) \):

\[ \text{Number of odd divisors} = (3 + 1)(2 + 1) = 4 \times 3 = 12 \]

### Conclusion

In both cases, the number of odd divisors of \( n \) is \( 12 \).

Thus, the final answer is:

\[ \text{The number of odd divisors of } n \text{ is } 12. \]

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