Consider the two number: 37 and 52
What is the remainder when we divide 37 by 7? 2 of course. And 52 produces remainder 3 when divided by 7. Suppose we want to know the remainder when the product of 37 and 52 is divided by 7.
One way to do this is to first multiply 37 and 52 to get 1924, and then divide it by 7 to get 274 as quotient and 6 as remainder. Indeed
However there is a simpler method to do this. If we just multiply the remainders produced by 37 and 52 we will get the final remainder! Indeed . Apparently if the numbers are multiplied that the remainders also get multiplied!
Let us do one more experiment. This time we divide by 9. Suppose the numbers are 83 and 904. 83 produced 2 as remainder ( ) and 904 produced 4 as remainder ( ) . Then what do we expect the remainder to be when is divided by 9? It should be the product of the individual remainders or . Indeed we find .
The question is why this happens? Let us approach the problem algebraically. Suppose be two numbers and m is the number by which we divided both them. Let the quotients and remainders produced be respectively. That is
Thus when is divided by n , quotient is and remainder is which is the product of the initial remainders. So it is no accident that if we multiply the initial remainders of two numbers we get the final remainder produced by the product of those two numbers.
However what will happen if exceeds n? Remainder cannot exceed the divisor. So we divide again by n to find the final remainder. That is suppose
then is the final remainder. Infact the final quotient and remainder will be formed in the following manner:
Try to verify this if the numbers are 48 and 54 and the divisor is 5.
This same logic works when two numbers are added or a number is raised to some power. In the next installment of this series of articles on number theory we will hand all of these operations in detail.