If $n = 2^{10} \cdot 3^{14} \cdot 5^{8}$, how many of the natural-number factors of $n$ are multiples of 150?
$150=2^13^15^2$.  Thus the coefficient of $2$ must be between $1$ and $10$, the coefficient of $3$ must be between $1$ and $14$, and the coefficient of $5$ must be between $2$ and $8$.  So the number of possible factors is

$$(10)(14)(7)=\boxed{980}$$
980
Level 5
Number Theory
Problem: If $n = 2^{10} \cdot 3^{14} \cdot 5^{8}$, how many of the natural-number factors of $n$ are multiples of 150?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine how many natural-number factors of \( n \) are multiples of 150, we first need to express 150 in terms of its prime factors. The prime factorization of 150 is \( 150 = 2^1 \cdot 3^1 \cdot 5^2 \).

Next, we analyze the prime factorization of \( n \):
\[
n = 2^{10} \cdot 3^{14} \cdot 5^{8}.
\]

A factor of \( n \) that is a multiple of 150 must include at least \( 2^1 \), \( 3^1 \), and \( 5^2 \). Therefore, we can express any factor of \( n \) that is a multiple of 150 in the form:
\[
f = 2^a \cdot 3^b \cdot 5^c,
\]
where \( a \geq 1 \), \( b \geq 1 \), and \( c \geq 2 \).

Next, we determine the possible values for \( a \), \( b \), and \( c \) based on the constraints from \( n \):
- For \( a \): Since \( a \) must be at least 1 and can go up to 10 (the exponent of 2 in \( n \)), the possible values for \( a \) are \( 1, 2, \ldots, 10 \). This gives us \( 10 - 1 + 1 = 10 \) options.
- For \( b \): Since \( b \) must be at least 1 and can go up to 14 (the exponent of 3 in \( n \)), the possible values for \( b \) are \( 1, 2, \ldots, 14 \). This gives us \( 14 - 1 + 1 = 14 \) options.
- For \( c \): Since \( c \) must be at least 2 and can go up to 8 (the exponent of 5 in \( n \)), the possible values for \( c \) are \( 2, 3, \ldots, 8 \). This gives us \( 8 - 2 + 1 = 7 \) options.

Now, we calculate the total number of factors of \( n \) that are multiples of 150 by multiplying the number of options for \( a \), \( b \), and \( c \):
\[
\text{Total factors} = (\text{options for } a) \times (\text{options for } b) \times (\text{options for } c) = 10 \times 14 \times 7.
\]

Calculating this gives:
\[
10 \times 14 = 140,
\]
and then,
\[
140 \times 7 = 980.
\]

Thus, the final answer is \( 980 \).<|END_OF_LATENT|> The final answer is $980$.