Karatsuba worked multiplication

Use decimal expansion, $B=10$, $n=4$, $m=2$, and $q=100$. The Karatsuba operand splitting is $1234=12\cdot 100+34$, $5678=56\cdot 100+78$. The Karatsuba three-product identity givesThe third product has a three-digit operand. A small-input base case can evaluate these three products directly, or a smaller threshold can continue recursion. Either choice preserves the identity; only the operation count changes.

The same Karatsuba operand splitting handles $123\cdot 45$: choose $n=3$, $m=2$, and write $123=1\cdot 100+23$, $45=0\cdot 100+45$. Then $p_2=0$, $p_0=1035$, $p_s=24\cdot 45=1080$, and $h=45$, giving $4500+1035=5535$. A zero high block and unequal lengths require no new algebraic rule.