Extending Foiling

The FOILing method can be extended to multiply two three-digit numbers by treating the last two digits of each number as a single unit. This effectively turns a three-digit multiplication into a more manageable two-digit multiplication problem.

For example, to multiply $abc \times def$, you can group them as $[a | bc] \times [d | ef]$. The steps are:

• Last: Multiply the two-digit units: $bc \times ef$. The last two digits of this product are the last two digits of your final answer. Keep track of the carry.
• Inner+Outer: Multiply the inner and outer pairs and add them, plus the carry: $(a \times ef) + (d \times bc) + carry$. The last two digits of this result are the next two digits of your answer. Again, track the new carry.
• First: Multiply the leading digits and add the final carry: $a \times d + carry$.

This method is particularly effective for squaring three-digit numbers, where memorizing squares of two-digit numbers is highly beneficial.