## (Math Quiz 10 – CT) How many numbers are positive divisors of ({2^{10}}{.3^6}{.5^8}) and divisible by ({2^5}{.3 ^2}{.5^4})?

• Question:

How many numbers are positive divisors of $${2^{10}}{.3^6}{.5^8}$$ and divisible by $${2^5}{.3^2}{. 5^4}$$?

Reference explanation:

Notice that $${2^{10}}{.3^6}{.5^8} = {2^5}{.3^2}{.5^4}\left( {{2^5) }{{.3}^4}{{.5}^4}} \right)$$
For every positive divisor of $${2^5}{.3^4}{.5^4}$$ when multiplied by $${2^{10}}{.3^6}{.5^8}$$ are all positive divisors of satisfying the requirements. The number of positive divisors to look for is: $$\left( {5 + 1} \right)\left( {4 + 1} \right)\left( {4 + 1} \right) = 150$$.