Multiplying Reverses

This trick applies to multiplying two two-digit numbers where the digits are reversed (e.g., $ab \times ba$).

The algebraic expansion is $(10a+b)(10b+a) = 100(ab) + 10(a^2+b^2) + ab$.

Procedure:

• Ones digit: The product of the two digits ($a \times b$).
• Tens digit: The sum of the squares of the digits ($a^2 + b^2$), plus any carry.
• Hundreds digit: The product of the two digits ($a \times b$), plus any carry.

For example, $53 \times 35$: Ones is $3 \times 5 = 15$ (write 5, carry 1). Tens is $3^2+5^2+1 = 9+25+1=35$ (write 5, carry 3). Hundreds is $3 \times 5 + 3 = 18$. The result is 1855.