The other day while biking I realized that if you multiply a single digit number by a multi digit number, and then add the digits together until you got to a single digit number, it would have the same final digit as if you added the multi digit number together before multiplying.
For example, $423 \cdot 7=2961, 2+9+6+1=18, 1+8=9$, while $(4+2+3) \cdot 7=63, 6+3=9$.
How would one go about writing a proof for this, or does one already exist? I'm not super into math, just like playing with numbers while I ride my bike, so thought this was interesting, and was curious as to the why of it.
