Remainders Theorem

Concept of Remainders

We know that when a number M is divided by another number N, and if M > N, then the remainder is calculated by subtracting the maximum possible multiple of N from M.

In the process the excess portion remained after subtraction is called remainder, the maximum multiple of N is called quotient, and M and N are called dividend and divisor respectively.

We have the following relation among them:

Dividend = Quotient x Divisor + Remainder

Important points on the remainder

• The remainder is always less than the divisor.
• If the remainder is 0, then the divisor is called the factor of the dividend
• If the dividend is less than the divisor, then the remainder is dividend itself.

E.g., 5 divided by 6, the remainder is 5 only.

• The remainder should always be calculated in its actual form. i.e. it should not be reduced into simplest form.

E.g., Remainder of when 4 is divided by 6 is 4 and NOT 2

The concept of the negative remainder

In mathematics, the remainder is always non-negative integers. However to we can use the concept of negative integers to solve many questions easily and with less calculation.

Let us take an example to understand it:

We know that remainder obtained when 15 is divided by 4 is 3 as 15 is 3 more than the nearest multiple of 4 i.e. 12.

Here we are comparing 15 with 12 (the largest multiple of 4 less than equal to 15 ) and found that 15 is 3 more than 12 and hence the excess portion is 3 which is the remainder.

Instead of comparing with 12, if we compare 15 with 16 (another multiple of 4 close to 15) we can say that 15 is 1 LESS than 16. Or there is a deficiency of 1 in 15 to make it divisible by 4. This deficient number is called a negative remainder. i.e. we can say that the remainder when 15 is divided by 4 is -1.

Conversion from negative remainder to the corresponding positive remainder and vice versa

Remainder obtained when N is divided by 5 is -2, then the positive remainder is given by 5-3 = 2. Similarly, if 4 is the positive remainder obtained when some number is divided by 7, then the negative remainder is given by 4-7 = -3.

We will be using this concept in solving few examples to understand the application of negative remainders.

Note: Product, addition and subtraction of any two or more numbers has the same remainder when divided by any natural number, as the corresponding product, addition and subtraction of their remainders.

Let us understand this with an example:

Take a number N =. Now let us find the remainder when N is divided by 5.

One approach is that we first calculate the value of N =i.e. which is equal to 197. And then dividing this number by 5 to get the remainder = 2.

The other approach is that we calculate the remainders of each number given in N by dividing it by 5 and then use the corresponding mathematical operators with these remainders.

Remainder obtained when 24 is divided by 5 = 4

Remainder obtained when 8 is divided by 5 = 3

Remainder obtained when 12 is divided by 5 = 2

Remainder obtained when 7 is divided by 5 = 2

Replacing numbers with their corresponding remainders we get,. Since 12 is greater than 5, we divided 12 by 5 again to get the final remainder = 2.

Note: The second approach might sound difficult or redundant for the given example. But the detailed explanations are given for the understanding purpose, we will see that how applying this second approach we can solve advanced problems.

Applying both positive remainder and the concept of negative remainder simultaneously in the above problem with the second approach:

• Remainder obtained when 24 is divided by 5 = -1
• Remainder obtained when 8 is divided by 5 = -2
• Remainder obtained when 12 is divided by 5 = 2
• Remainder obtained when 7 is divided by 5 = 2
• Replacing numbers with their corresponding remainders we get,. Hence the final remainder = 2.

Questions on remainders

Question: What is the remainder when 123 × 124 × 125 is divided by 9.

Solution

Remainder obtained when 123 is divided by 9 = -3

Remainder obtained when 124 is divided by 9 = -2

Remainder obtained when 123 is divided by 9 = -1

Final remainder = (-3)(-2)(-1) = -6. The required positive remainder = 9-6 = 3.

Question: What is the remainder when 1! + 2! + 3! + …. + 100! Is divided by 5.

Solution

Observe that in the series 5! onwards every number is divisible by 5 i.e. the remainder in each case is 0.

So the required remainder is obtained by dividing only the first 4 numbers i.e.

Question: What is the remainder when  divided by 17. 

Solution

This question can be solved without doing any calculation by applying Fermat Theorem, which is the topic in the next section.

We will solve this question by applying the concepts we learned here.

Observe that 24 =16 and 

Let us reduce the problem using this negative remainder.

i.e.