Finding Remainder Easy Part - 1

 

Finding Remainder

Do you think finding remainder is easy?

Yes, if you are dividing one big number by smaller.

But consider a situation where you find yourself perplexed and helpless and you don’t have any idea to calculate the remainder because you can’t apply the basic rule –

Dividend = Divisor × Quotient + Remainder

Let’s first begin with Euclid Division lemma

As you know that Euclid is called the father of geometry but he has contributed immensely in Arithmetic and Number theory also.

Euclid Division Lemma

Let a and b be any two positive integers then there exist unique integers q and r such that a = bq + r where 0 ≤ r < b

This is basically the Euclid Division Lemma

                          

In the first example Dividend = 445, Divisor = 17,   Quotient = 26 and Remainder = 3

Apply the division lemma

Dividend = Divisor × Quotient + Remainder

          445 = 17 × 26 + 3

                 = 442 + 3

Here, Remainder < Divisor

In the second example, remainder = 0 which also satisfies the division lemma. Let’s check

Dividend = Divisor × Quotient + Remainder

2412 = 36 × 67 + 0

2412 = 2412

But what will be the remainder when 2 is divided by 6?

Remember, if numerator is less than denominator, i.e. when dividend is less than divisor then dividend will be remainder.

Here, obviously, 2 is the remainder

Some people write 2 ÷ 6 as and further cancel out to get and write 1 as quotient which is wrong. Finding remainder is something different from simplifying fraction or changing a fraction into simplest form. Therefore, you are advised not to cancel the number.

In the above two examples we have seen the remainder is a positive integer but in finding the remainder of larger number the concept of negative integer remainder also plays a big role.

Negative Remainder

Let’s take an example

26 = 9 × 2 + 8

26 = 9 × 3 – 1

In the first case remainder = + 8 while in the second case remainder = –1.

Negative remainder reduces the effort of calculation.

54 = 11 × 4 + 10                                             Remainder = + 10

54 = 11 × 5 – 1                                                Remainder = – 1

 

Finding remainder of product

Example: - Find the remainder when 33 x 34 x 35 is divided by 8

Solution: - The first thing we can do it to multiply 33, 34 and 35

                        33 × 34 × 35 = 39270

Now divide 39270 by 8 . What do you get?

Remainder = 6

Let’s try it by remainder method.

On division of 33 by 8 we get remainder = 1

On division of 34 by 8 we get remainder = 2

On division of 35 by 8 we get remainder = 3

Hence  Remaider = 6

The conclusion that we can draw from the above example is that if a, b, c ---- be numbers divided by w gives remainder x, y, z - - respectively then the product of a, b, c - - - when divided by the same divisor w will give the same remainder obtained by dividing the product of remainder x, y, z --- by w.

First divide 121, 118 and 124 by 60 to get remainders as 1, –2 and 4 respectively.

Therefore final remainder can be obtained without actual multiplication of 121× 118 × 124 and it will be the product of individual remainder of numbers and i.e.  (1×–2×4) = –8

Final remainder = –8 + 60 = 52.

Example: Find the remainder when 37⁵⁰⁰ is divided by 9

Solution:- As we know that 3⁴ = 3 × 3 × 3 × 3 = 81

Hence, 37⁵⁰⁰ = 37 × 37 × - - - 500 times.

Multiplying 37 five hundred times is a big deal so we need to think something extra ordinary. We divide 37 by 9 and get the remainder 1

Therefore, 37⁵⁰⁰  divided by 9 is equal to 1⁵⁰⁰divided by 9

Hence, Remainder = 1

Example: Find the remainder when 54 × 63 × 86 is divided by 11

Solution:-

On division of 54 by 11 we get remainder = 10

On division of 63 by 11 we get remainder = 8

On division of 86 by 11 we get remainder = 9

Through mental calculation we can find the remainder on dividing 720 by 11.

720 = 11 × 65 + 5; Remainder = 5

Let’s apply the concept of negative integer

On division of 54 by 11 we get remainder = – 1 because 54 – 5 × 11 = – 1

On division of 63 by 11 we get remainder = – 3 because 63 – 6 × 11 = –3

On division of 86 by 11 we get remainder = – 2 because 86 – 8 × 11 = – 2

Since final remainder can’t be – 6, so we add the divisor 11 to make the remainder positive at the end.

Remainder = – 6 + 11 = 5

This shows that the negative remainder concept saves our time.

DR Rajesh Thakur