## geerky42 3 years ago How many numbers in the form $$a^4$$, where $$a \in \mathbb{Z}^+$$ divide $$3! \times 4! \times 7!$$ ?

1. satellite73

i don't think there are too many since you need primes to the power of 4

2. satellite73

included in $$3!4!7!$$ is $$2^7$$ and $$3^4$$ all other primes are to lower powers

3. satellite73

i am not certain but on the basis of prime factorization i only see $2^4,3^4,(2\times 3)^4$

4. satellite73

oops i miscounted!! it is $$2^8$$ and $$3^4$$

5. satellite73

so maybe there are 4 all together, $2^4, 3^4, (2^2)^4,(2\times 3)^4, (2^2\times 3)^4$

6. satellite73

well that is actually 5, not 4