### Instruction: The base-10 numbers 217 and 45 are multiplied. The product is then written in base-6. What is the units digit of the base-6 representation?### Response: I need to find the base-6 representation of 217 * 45.
One way to do this is to convert both numbers to base-6 first, and then use the base-6 multiplication table.
But that might be tedious and prone to errors.
Another way is to multiply them in base-10 first, and then convert the result to base-6.
That seems easier, since I am more familiar with base-10 multiplication.
So I will do 217 * 45 in base-10.
I can use the standard algorithm of multiplying each digit and adding the partial products.
I get 217 * 45 = 9765.
Now I need to convert 9765 to base-6.
To do this, I can repeatedly divide by 6 and write down the remainders.
The remainders will be the digits of the base-6 representation, starting from the right.
So I start with 9765 / 6 = 1627 with a remainder of 3.
That means the units digit of the base-6 representation is 3.