Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.

Remainder

Age 11 to 14 Challenge Level:

What is the remainder of $2^{2002}\div 7?$

What happens with different powers of $2$?

Try to explain the mathematics behind what you discover.