trick, but I do not understand the theory that allows the final step of the

solution to be valid.

Someone selects two cards and given the cards are '5' and '3,' the

following equation can be reached:

'10x + y = 53'

Dividing 53 by 10, the quotient is 5 and the remainder is 3, but I do not

understand what theory would make this step valid. I have considered simple

division and mod-ing, but I still don't understand what theory or property

makes this possible.

With the equation:

'12x + y = 38,'

dividing 38 by 12, the quotient is 3 and the remainder is 2, which are the

values of the two cards selected in another manipulation of the same card

trick. I believe this has something to do with varying the base of the

equation, but I'm still unsure of what makes this valid or if it is even

correct.

Thank you for any help.