You meet Sparky, and as usual he throws you a mathematical challenge… “Take any three digit number and repeat it to form a six-digit number.”

Sparky helpfully warns that to make a true three digit number, the first digit can not be zero. “For example, $345$ becomes $345345$ and $558$ becomes $558558$, but $012$ becomes $12012$, which ain’t a six-digit number!”

“Good point,” you concede as you wonder where this is headed.

Soon Sparky commands you to “divide your six-digit number by $7$, then divide the result by $11$, and finally divide that result by $13$.”

As you calculate Sparky boldly states, “No matter what number you start with not one of your division problems will result in a remainder. Ever!”

Is Sparky correct, and if so, why?

But that’s not all. Rising to his moment, Sparky jumps onto the table and triumphantly proclaims, “Once you complete all three division problems, the result will always be the exact same three-digit number that you started with!”

Is Sparky correct, and if so, why?