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
