Catherine rolls a standard $6$-sided die six times. If the product of her rolls is $100,$ then how many different sequences of rolls could there have been? (The order of the rolls matters.)
The prime factorization of $100$ is $2^2\cdot 5^2.$ Since the prime factorization of an ordered $6$-tuple of numbers must have an exponent that is a multiple of $6,$ the exponent of $2$ in the prime factorization of an ordered $6$-tuple of numbers whose product is $100$ must be $0,$ $6,$ $12,$ $18,$ or $24.$
Since the exponent of $2$ can be $0,$ $6,$ $12,$ $18,$ or $24$ and the exponent of $2$ in $100$ is $2,$ there are $5$ positions for the $2$'s in the exponent of $2.$ Therefore, the number of different sequences of rolls is $5\cdot 1\cdot 1\cdot 1\cdot 1\cdot 1 = \boxed{5}.$