### Instruction: What is the product of the least common multiple and the greatest common factor of $20$ and $90$?### Response: I think we should start by finding the LCM and the GCF of 20 and 90.
I agree. I'll start by finding the LCM. The prime factorization of 20 is $2^2 * 5$ and the prime factorization of 90 is $2 * 3^2 * 5$.
So the LCM is $2^2 * 3^2 * 5$?
Yes, that is correct.
I'll start on the GCF then. The prime factorization of 20 is $2^2 * 5$ and the prime factorization of 90 is $2 * 3^2 * 5$.
The GCF is the product of the shared prime factors, so it is $2 * 5$.
So the product of the LCM and the GCF is $(2^2 * 3^2 * 5) * (2 * 5)$?
Yes, that is correct.
We can simplify that to $2^3 * 3^2 * 5^2$.