 What if we multiply a base n number by more than n? So, for example, 5 times 1, 3, 2, base 4. Now, remember, in base n, the largest number you can write is n, but a problem exists whether or not you can solve it. So let's think about this. 5, 1, 3, 2, base 4 is really 4, 1, 3, 2, base 4s, and one more, 1, 3, 2, base 4. And we can generalize this process. Suppose you have a copies of n to which you add b copies of n, then you have a plus b copies of n. This means that a times n plus b times n equals a plus b times n. So let's multiply 7 times 2 and 3, base 5, and let's also rewrite the multiplication. Now, since we're working base 5, the largest amount we can talk about is 5, because remember in base n, the largest number you can write is n. Now, while you understand the concept 7, pretend you didn't. So, what is 7? This is really 5 and 2. It could be other things as well, but we'll use 5 and 2. So 7, 2, 1, 3, base 5s would be 5, 2, 1, 3, base 5s, and 2 more, 2, 1, 3, base 5s. Now, the 5, 2, 1, 3, base 5s would give us 2, 1, 3, 0, base 5. Remember, when you multiply base n number by n, all digits shift one place. 2 more 2, 1, 3, base 5s would give us 3, 1, 1, base 5. And note that 7 can be expressed in base 5 as 1, 2, base 5. And so we can rewrite the product. 1, 2, base 5 times 2, 1, 3, base 5 equals 3, 1, 1, 1, base 5. And we can work in the other direction. Let's say we want to multiply 1, 3, base 4 by 2, 3, base 4. So note that 1, 3, base 4 is 1, 4, and 3. So 4, 2, 3, base 4s will be 2, 3, 0, base 4. 3 more 2, 3, base 4s will be 2, 0, 1, base 4. And if we put these all together, we'll have 1, 0, 3, 1, base 4. And so 1, 3, base 4 times 2, 3, base 4 equals 1, 0, 3, 1, base 4. Test 1, 2, 3, 4, 16, 24, 38, 56, 158.