What is the smallest number of marbles that could be divided up either into bags of $18$ marbles or into bags of $42$ marbles, with no marbles left over in each case?
I want to find the smallest positive integer that is a multiple of both $18$ and $42$. One way to do this is to find the least common multiple of $18$ and $42$. To find the least common multiple, I can use the prime factorization method. I write $18$ and $42$ as products of prime factors: $18 = 2 \cdot 3 \cdot 3$ and $42 = 2 \cdot 3 \cdot 7$. Then, I take the highest power of each prime factor that appears in either factorization and multiply them together: $2^1 \cdot 3^2 \cdot 7^1 = 126$. This means that $126$ is the least common multiple of $18$ and $42$, and the smallest number of marbles that could be divided up either into bags of $18$ marbles or into bags of $42$ marbles, with no marbles left over in each case. 