Write on the blackboard the lucky dice numbers 7 and 11, and the unlucky number 13.   Ask a student to write any three digit number on the blackboard, then repeat the digits to make a six digit number.
Ask the students to divide that number by 7.   You predict that there will be no remainder.   Ask them to divide this by 11.   Again, no remainder.   Now ask them to divide this last quotient by 13.   For the third time, there is no remainder, but now there is a bigger surprise. The result is the original three digit number!

Why does this work?

Let ABC be the three digit number that was chosen.
So the six digit number is of the form ABCABC.
ABC times 1,001 produces ABCABC.
Now the prime divisors of 1,001 are 7, 11, and 13,
so 7 x 11 x 13 = 1001.
If (7 x 11 x 13) x ABC = ABCABC,
then when ABCABC is divided by 7, 11, and 13,
the quotient will be ABC.

Here is a follow-up activity for this trick.

(1) Pick any three digit number (e.g., 273).

(2) Then make a 12 digit number by repeating it three times
      (e.g., 273,273,273,273).

(3) Divide that number by 7.

(4) Divide that quotient by 11.

(5) Divide that quotient by 13.

What is your final quotient? Explain.

In this example, you would get 273,000,273.
The number is 273 x 1,000,001.