**Submitted by Sav2718:**

A part of a big list of cool number facts I’ve posted a few months ago (https://www.facebook.com/notes/gur-keren/cool-number-facts/590025287678630), many of which I discovered by myself and as far as I know I’m the only source of them. I chose this example (although can be generalized) because of its somewhat symmetrical nature. I call them “Train numbers”, “clone numbers” or “Gur numbers” (after myself ) .

Let’s have a look:

We claim that 166..6 (N 6s) times 4 is 66..64 (N 6s) (the exact same process of proof is true for our second example of 19,199,1999… times 5).

For N=1:

16*4=64 , true

Assume our claim is true for N=K-1 so lets have a look at N=K:

166..66 (K 6s) * 4=(166..66 [K-1 6s] * 10+6)*4=166..66 [K-1 6s]*10*4+6*4=(166..66 [K-1 6s]*4)*10+24 but we assumed our claim is true for K-1 6s so:

(166..66 [K-1 6s]*4)*10+24=(66..664 [K-1 6s])*10+24=66..6640+24 [K-1 6s]= 66..6664 [K 6s] so our assumption was correct.