FOILing/LIOFing When Multiplying

A fast multiplication method for two-digit numbers, $ab$ and $cd$, is LIOFing (Last-Inner+Outer-First). This is derived from the algebraic expansion $(10a + b)(10c + d) = 100(ac) + 10(ad + bc) + bd$.

• Last: The ones digit is $b \times d$.
• Inner+Outer: The tens digit is $(a \times d) + (b \times c)$.
• First: The hundreds digit is $a \times c$.
• Remember to handle carries at each step. For example, $74 \times 23$: Last is $4 \times 3 = 12$ (write 2, carry 1). Inner+Outer is $(7 \times 3) + (2 \times 4) + 1_{carry} = 21 + 8 + 1 = 30$ (write 0, carry 3). First is $(7 \times 2) + 3_{carry} = 14 + 3 = 17$. The result is 1702.