Remainder Theory

Remainder Theory

Questions from Number System appear regularly in almost all competitive exams. Within number system, the questions on remainders are found to be most tricky. This article will help you learn different types of remainder questions and various approaches you can apply to solve these.

The basic remainder formula is:

• Dividend = Divisor × Quotient + Remainder

If remainder = 0, then it the number is perfectly divisible by divisor and divisor is a factor of the number e.g. when 8 divides 40, the remainder is 0, it can be said that 8 is a factor of 40.

You will understand the concept better with the help of the following examples:

Example 1:

Find the Remainder; [(12 × 13 × 14)/5]

Solution:

Remainder [(12 × 13 × 14)/5] = Remainder [2184/5] = 4. But is it the right method? Instead, find the remainder for each term when divided by 5, and replace each term with the respective remainders. Remainder (12 × 13 × 14)/5 is same as Remainder (2 × 3 × 4)/5, i.e. 24/5 = 4.

In this case,12,13 and 14 will give remainders 2, 3 and 4, respectively when divided by 5. So, replace them with the respective remainders in the expression and find the remainder.

Note: One common mistake while dealing with remainders is that, when the numbers have common factors in both dividend and divisor e.g. what is the remainder when 15 is divided by 9. Can it be solved as- 15/9 is same as 5/3.

Hence, remainder is 2. No! 15/9 will give a remainder of 6. Always remember that if you find the remainder after cancelling common term, make sure you multiply the remainder obtained with the common factors removed. In this case, you will get correct answer (6) when you multiply the remainder obtained i.e. 2 with the common factor you cancelled i.e. 3.

There are few important results relating to numbers. Those will be covered one by one in the following examples.

1. Formulas Based Concepts for Remainder:

• (aⁿ + bⁿ) is divisible by (a + b), when n is odd.
• (aⁿ - bⁿ) is divisible by (a + b), when n is even.
• (aⁿ - bⁿ) is always divisible by (a - b), for every n.

Example 2:

What is the remainder when 15³¹ + 23³¹ is divided by 19?

Solution:

15³¹ + 23³¹ is divisible by 15 + 23 = 38 (as 31 is odd). So, the remainder will be 0.

2. Concept of Negative Remainder:

By definition, remainder cannot be negative. But in certain cases, you can assume that for your convenience. But a negative remainder in real sense means that you need to add the divisor in the negative remainder to find the real remainder.

Example 2:

What is the remainder when 2¹¹ is divided by 3?

Solution:

The easiest method to solve this problem is by using the concept of negative remainders. Here, 2 when divided by 3gives a remainder of -1, which is theoretically incorrect but can be taken for the sake of convenience.

You are asked to find (-1) × (-1) ×…11 times divided by 3. Now, the product of  -1, when taken 11 times will give a product of – 1 only, which in real sense will give a remainder of – 1 + 3 = 2.

NOTE: Whenever you are getting a negative number as the remainder, make it positive by adding the divisor to the negative remainder.

3. Cyclicity in Remainders:

Cyclicity is the property of remainders, due to which they start repeating themselves after a certain point. Learn how to apply this concept with the help of following example.

Example 3:

What is the remainder when 2⁵¹ is divided by 7?

Solution:

Remainder (2¹/7) = 2; Remainder (2²/7) = 4; Remainder (2³/7) = 1; Remainder (2⁴/7) = 2.

As you can see, the remainders start repeating after the first three steps i.e. a cycle of 2, 4 and 1 is present in the remainders. So, at power 51 the remainder will be same as on power 3.

Hence, Remainder (2⁵¹/7) = 1.

4. Role of Euler’s Number in Remainders:

Euler’s Remainder theorem states that, for co-prime numbers M and N, Remainder [M^E(N) / N] = 1, i.e. number M raised to Euler number of N will leave a remainder 1 when divided by N. Always check whether the numbers are co-primes are not as Euler’s theorem is applicable only for co-prime numbers.

Example 4:

What is the remainder when 21²⁵⁶ is divided by 17?

Solution:

Remainder [21/17] = 4

Remainder [21²⁵⁶/17] = Remainder [4²⁵⁶/17]

4 and 17 are co-prime numbers and E (17) = 17 × [1 – (1/17)] = 16.

So, Euler’s theorem says Remainder [4¹⁶/17] = 1

Now, 4²⁵⁶ can be written as 4^16*16

Remainder [4²⁵⁶/17] = Remainder [4^16*16/17] = Remainder [(4¹⁶)¹⁶/17] = Remainder [(1)¹⁶/17] = 1.

With the application of Euler’s number, you can make almost 90% of the remainder questions easy and solve those in a relatively lesser time period.