### Instruction: Find the product of the greatest common divisor and the least common multiple of $18$ and $42.$.### Response: To find the product of the GCD and the LCM of $18$ and $42,$ I need to find the GCD and the LCM first.
One way to find the GCD and the LCM is to use prime factorization.
I can write $18$ and $42$ as products of prime factors: $18 = 2 \times 3 \times 3$ and $42 = 2 \times 3 \times 7.$
To find the GCD, I need to find the common prime factors with the lowest powers: $GCD(18,42) = 2 \times 3 = 6.$
To find the LCM, I need to find the prime factors with the highest powers: $LCM(18,42) = 2 \times 3 \times 3 \times 7 = 126.$
Now I have the GCD and the LCM, and I can find their product: $GCD(18,42) \times LCM(18,42) = 6 \times 126 = 756.$