Key Concepts and Formulas

• Prime factorization of a number.
• Number of divisors of a number $n = p_1^{a_1} p_2^{a_2} ... p_k^{a_k}$ is given by $(a_1 + 1)(a_2 + 1)...(a_k + 1)$.
• Solving simultaneous equations.

Step-by-Step Solution

Step 1: Solve for y and z

We are given the equations:

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

From equation (2), we can write:

$\frac{y + z}{yz} = \frac{5}{6}$

Substituting $y + z = 5$ from equation (1):

$\frac{5}{yz} = \frac{5}{6}$

$yz = 6 \quad ...(3)$

Step 2: Find the values of y and z

We now have two equations:

$y + z = 5$ $yz = 6$

We can use the identity $(y - z)^2 = (y + z)^2 - 4yz$ to find $y - z$:

$(y - z)^2 = (5)^2 - 4(6) = 25 - 24 = 1$

$y - z = \pm 1$

Since we are given that $y > z$, we take the positive root:

$y - z = 1 \quad ...(4)$

Now we have a system of linear equations:

$y + z = 5$ $y - z = 1$

Adding the two equations:

$2y = 6$ $y = 3$

Substituting $y = 3$ into $y + z = 5$:

$3 + z = 5$ $z = 2$

Therefore, $y = 3$ and $z = 2$.

Step 3: Determine the prime factorization of n

We are given that $n = 2^x 3^y 5^z$. Substituting the values of $y$ and $z$ we found:

$n = 2^x 3^3 5^2$

Step 4: Calculate the number of odd divisors

We want to find the number of odd divisors of $n$. Odd divisors will not have any factor of 2. Thus we consider only the powers of 3 and 5 in the prime factorization. So we are looking for the number of divisors of $3^3 5^2$. The number of divisors is given by $(3 + 1)(2 + 1) = 4 \times 3 = 12$.

Therefore, the number of odd divisors of $n$ is 12.

Common Mistakes & Tips

• Remember to only consider the powers of odd prime factors when calculating the number of odd divisors.
• Double-check your algebra when solving simultaneous equations.
• Always read the question carefully to understand what is being asked.

Summary

We first solved for $y$ and $z$ using the given equations. We found $y = 3$ and $z = 2$. Then, we substituted these values into the prime factorization of $n$ to get $n = 2^x 3^3 5^2$. Finally, we calculated the number of odd divisors of $n$ by considering only the powers of 3 and 5, which resulted in $(3+1)(2+1) = 12$.

Final Answer

The final answer is \boxed{12}, which corresponds to option (C).