The topic of remainders is an important one for the CAT exam. The questions from this topic often appear in CAT or other competitive exams. In this article we are going to discuss this topic in detail.

We know that Dividend = Divisor × quotient + remainder

The number which is to be divided is known as dividend and the number which divides the dividend is known as divisor. Let us take an example:

If we divide 23 by 5, then 23 is dividend and 5 is divisor. Now 23 = 5 × 4 + 3, so, 4 is the quotient and 3 is the remainder. Similarly, if 99 is divided by 7, then 99 = 7 × 14 + 1. So, 14 is the quotient and 1 is the remainder.

Negative Remainders: Generally, the remainder is always a non negative number but sometimes to solve the questions in an easy way, we can take the remainders as negative also. Negative remainders are simply the difference of the positive remainder and the divisor and the sign used is negative. Let us try to understand it with the help of an example. If we divide 23 by 5, the positive remainder is 3. Now the difference of 3 and 5 is 2, so the negative remainder when 23 is divided by 5 is – 2. Similarly, when 99 is divided by 7, the remainder is 1 or – 6.

Note: The negative remainder can be converted into the positive remainder by adding divisor to it.

Few Basic Rules to find the remainders when sum, product or the difference of the numbers is given:

I: When the sum of numbers is divided by same divisor D:

Let us consider that x/D gives remainder r₁ and y/D gives remainder r₂. Note that here the divisor is same in both the cases.

Now if we divide (x + y) by the same divisor D, the remainders will also get added. So, the final remainder will be r₁ + r₂. Three cases will arise here:

Case I: If r₁ + r₂ < D, then r₁ + r₂ will be the remainder

Case II: If r₁ + r₂ = D, then the remainder will be zero

Case III: If r₁ + r₂ > D, then divide r₁ + r₂ by D to get the final remainder.

II: When the product of numbers is divided by same divisor D:

Let us consider that x/D gives remainder r₁ and y/D gives remainder r₂. Note that here the divisor is same in both the cases.

Now if we divide (x × y) by the same divisor D, the remainders will also get multiplied. So, the final remainder will be r₁ × r₂. Three cases will arise here:

Case I: If r₁ × r₂ < D, then r₁ × r₂ will be the remainder

Case II: If r₁ × r₂ = D, then the remainder will be zero

Case III: If r₁ × r₂ > D, then divide r₁ × r₂ by D to get the final remainder.

Note: In the above discussion, we have taken two numbers x and y only, but this result is applicable for ‘n’ number of numbers. So, if we get ‘n’ different remainders by dividing ‘n’ different numbers with the same divisor D, then on dividing the sum/product of all those numbers with the same divisor D, the remainders will get added/multiplied and the above three cases will be applicable again.

III: When the difference of numbers is divided by same divisor D:

Let us consider that x/D gives remainder r₁ and y/D gives remainder r₂. Note that here the divisor is same in both the cases.

Now if we divide (x – y) by the same divisor D, the remainders will also get subtracted. So, the final remainder will be r₁ – r₂. Now in this case, if the difference is positive, then that will be our remainder and if the difference of r₁ and r₂ comes out to be negative then we convert that negative difference or negative remainder to the positive remainder by adding the divisor to it.