Remainders in math expressions

A Remainder is the portion is left unused after doing some process,In this context remainder is a number which is left when a number is divided by another number.

In the basic levels of our education,we learn how to find remainders of normal maths expressions like Remainder("5",4)=1 but what if the question is asked in the form of Remainder( "19*19*17*21*9" ,"5") calculation part requires more time than the time we get to solve the same in the exam if we follow traditional methods.

Divisor A number by which another number to be divided.

Dividend: A number to be divided by another number.

Here is the trick which may help you to save your valuable time in competitive exams to solve these type of exams in less time.

Algorithm:

1. Split the numbers into its factors (you may also use prime factorization)
2. Repeat step 3 and 4 until remainder becomes less than quotient.
3. Now divide each and every number by the divisor and note the remainders.
4. Multiply the remainders and divide it by the divisor.
5. Display Remainder.

*********************************************************************************

Example 1: What is the remainder when 127*129*131 is divided by 4?

solution:

Remainder= Remainder("127",4)* Remainder("129",4)* Remainder("131",4)

Remainder=3*1*3=9(9>4 so divide by 4)

Remainder=Remainder("9",4)=1

Example2: What is the remainder when (123)^123 is divided by 7?

solution:

It can be split as 123*123*123........................(123 times)

Divide 123 by 7 (remainder: 4)

Remainder=(4)^123

Now adjust the divisor as 4* (4*4)^61                   // [1+2*61]

Remainder=4* Remainder("16",7)^61=4* (2)^61

Adjust the divisor again as  4* 2* (2*2*2)^20       //[1+3*20]

Remainder=4* 2* Remainder("8",7)=4*2*1

Remainder=8(8>7 so divide by 7 again)

Remainder=1

Feel free to comment if you face any difficulty in solving any kind of problems.