Number Systems - Finding Remainders, Number Theory, Quantitative Aptitude Quant Notes | EduRev

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FINDING REMAINDERS OF A PRODUCT (DERIVATIVE OF REMAINDER THEOREM)
(i) If ‘a1’ is divided by ‘n’, the remainder is ‘r1’ and if ‘a2’ is divided by ‘n’, the remainder is r2. Then,
If a1+a2 is divided by n, the remainder will be r1 + r2.
If a1 - a2 is divided by n, the remainder will be r1 - r2. 
If a1 × a2 is divided by n, the remainder will be r1 × r2.


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.
Ex. If 21 is divided by 5, the remainder is 1 and if 12 is divided by 5, the remainder is 2 Then if (21 + 12 = 33) is divided by 5, the remainder will be (1 + 2 = 3).
If (21 - 12 = 9) is divided by 5, the remainder will be 1 - 2 = - 1.
But if the divisor is 5, - 1 is nothing but 4 (9 = 5 × 1 + 4)
So, if 9 is divided by 5, the remainder is 4 and 9 can be written as 9 = 5 × 2 - 1.
So here - 1 is the remainder. So - 1 is equivalent to 4 if the divisor is 5. Similarly - 2 is equivalent to 3.
If (21 × 12 = 252) is divisible by 5, the remainder will be (1 × 2 = 2).

(ii) If two numbers ‘a1’ and ‘a2‘ are exactly divisible by n. Then their sum, difference and product is also exactly divisible by n.
i.e., If ‘a1’ and ‘a2’ are divisible by n, then
a1 + a2 is also divisible by n
a1 - a2 is also divisible by n
and a1 × a2 is also divisible by n.

Example: 12 is divisible by 3 and 21 is also divisible by 3
Sol. So, 12 + 21 = 33, 12 - 21 = - 9 and 12 × 21 = 252 all are divisible by 3.


FINDING REMAINDERS OF POWERS WITH THE HELP OF REMAINDER THEOREM:

1. Example: What is the remainder if 725 is divided by 6?
Sol. If 7 is divided by 6, the remainder is 1. So if 725 is divided by 6, the remainder is 1 (because 725 = 7 × 7 × 7… 25 times. So remainder = 1 × 1 × 1…. 25 times = 125).

2. Example: What is the remainder, if 363 is divided by 14.
Sol. If 33 is divided by 14, the remainder is - 1. So 363 can be written as (33)21.
So the remainder is (- 1)21 = - 1.
If the divisor is 14, the remainder - 1 means 13. (14 - 1 = 13) by pattern method.

3. Example: Find remainder when 433 is divided by 7.
Sol. If 41 is divided by 7, the remainder is 4. (41 = 4 = 7 × 0 + 4)

If 42 is divided by 7, the remainder is 2 (42 = 16 = 7 × 2 + 2)

If 43 is divided by 7, the remainder is 1 [43 = 42 × 4, So Remainder = 1 (4× 2 = 8)]

If 44 is divided by 7, the remainder is 4 (44 = 43 × 4, so the Remainder = 1× 4 = 4)

Number Systems - Finding Remainders, Number Theory, Quantitative Aptitude Quant Notes | EduRev

The remainders of the powers of 4 repeats after every 3rd power.
So, as in the case of finding the last digit, since the remainders are repeating after every 3rd power, the remainder of 433 is equal to the remainder of 43 ( since 33 is exact multiple of 3) = 1.
(OR)
If 433 is divided by 7, the remainder is 1. So 433 = (43)11 is divided by 7, the remainder is 111 = 1.

APPLICATION OF BINOMIAL THEOREM IN FINDING REMAINDERS

The binomial expansion of any expression of the form

Number Systems - Finding Remainders, Number Theory, Quantitative Aptitude Quant Notes | EduRev

Number Systems - Finding Remainders, Number Theory, Quantitative Aptitude Quant Notes | EduRev
There are some fundamental conclusions that are helpful if remembered, i.e.
(a) There are (n + 1) terms.
(b) The first term of the expansion has only a.
(c) The last term of the expansion has only b.
(d) All the other (n - 1) terms contain both a and b.
(e) If (a + b)n is divided by a, then the remainder will be bn such that bn < a.

 

1. Example: What is the remainder if 725 is divided by 6?

Sol. 
(7)25 can be written (6 + 1)25. So, in the binomial expansion, all the first 25 terms will have 6 in it. The 26th term is (1)25. Hence, the expansion can be written 6x + 1. 6x denotes the sum of all the first 25 terms.

Since each of them is divisible by 6, their sum is also divisible by 6, and therefore, can be written 6x, where x is any natural number. So, 6x + 1 when divided by 6 leaves the remainder 1. 
(OR)

When 7 divided by 6, the remainder is 1. So when 725 is divided by 6, the remainder will be 125 = 1.

2. Example: Remainder when 2510 is divided by 576?
Sol. We need to find out the remainder of 2510 when divided by 576.
Please note that 576 = 242
There are couple of methods of solving this. 

Using Binomial Theorem

2510 = (24 + 1)10
In the expansion, there will be 11 terms where the powers of 24 will vary from 0 to 10.
If the power of 24 is greater than or equal to 2 in a term, that term will be divisible by 576
The terms that will not be divisible by 576 are the terms that have powers of 24 as 0 or 1.
Those terms are
10C1.241.19 + 10C0.240.110
= 10.24.1 + 1.1.1
= 241

So, Remainder is [2510/576] = 241

Wilson’s Theorem

If n is a prime number, (n - 1)! + 1 is divisible by n.

Lets take n = 5

Then (n - 1)! + 1 = 4! + 1 = 24 + 1 = 25 which is divisible by 5.

Similarly 
If n = 7

(n - 1)! + 1 = 6! + 1 = 720 + 1 = 721 which is divisible by 7.

Corollary
If (2p + 1) is a prime number (p!)2 + (- 1)p is divisible by 2p + 1.

For Example
If p = 3, 2p + 1 = 7 is a prime number
(p!)+ (- 1)p = (3!)2 + (- 1)3 = 36 - 1 = 35 is divisible by (2p + 1) = 7.


PROPERTY
If “a” is natural number and P is prime number then (ap - a) is divisible by P.
e.g. If 231 is divided by 31 what is the remainder?

Number Systems - Finding Remainders, Number Theory, Quantitative Aptitude Quant Notes | EduRev 
So remainder = 2

FERMAT’S THEOREM
If p is a prime number and N is prime to p, then Np -1 - 1 is a multiple of p.

Corollary
Since p is prime, p - 1 is an even number except when p = 2.
Therefore ( Number Systems - Finding Remainders, Number Theory, Quantitative Aptitude Quant Notes | EduRev) = M(p).

Hence either Number Systems - Finding Remainders, Number Theory, Quantitative Aptitude Quant Notes | EduRev -1 is a multiple of p, that is Number Systems - Finding Remainders, Number Theory, Quantitative Aptitude Quant Notes | EduRev = Kp ± 1, where, K is some positive integer.


BASE RULE AND CONVERSION
This system utilizes only two digits namely 0 & 1 i.e. the base of a binary number system is two.  E.g. 11012 is a binary number, to find the decimal value of the binary number, powers of 2 are used as weights in a binary system and is as follows:
1 × 23 = 8
1 × 22 = 4
0 × 21 = 0
1 × 20 = 1
Thus, the decimal value of 11012 is 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 = 13.


CONVERSION FROM DECIMAL TO OTHER BASES
We will study only four types of Base systems,
1. Binary system (0, 1)
2. Octal system (0, 1, 2, 3, 4, 5, 6, 7).
3. Decimal system (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
4. Hexa-decimal system (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C D, E, F) where A = 10, B = 11 … ,F= 15.
Let us understand the procedure with the help of an example
1. Example: Convert 35710 to the corresponding binary number.
Sol. To do this conversion, you need to divide repeatedly by 2, keeping track of the remainders as you go. Watch below:
As you can see, after dividing repeatedly by 2, we end up with these remainders:

Number Systems - Finding Remainders, Number Theory, Quantitative Aptitude Quant Notes | EduRev
These remainders tell us what the binary number is. Read the numbers outside the division block, starting from bottom and wrapping your way around the right-hand side and moving upwards.
(357)10 convert to (101100101)2.
This method of conversion will work for converting to any non-decimal base. Just don't forget to include the first digit on the left corner, which is an indicator of the base. You can convert from base-ten (decimal) to any other base.


CONVERSION FROM OTHER BASES TO DECIMAL
We write a number in decimal base as
345 = 300 + 40 + 5 = 3 × 102 + 4 × 101 + 5 × 100
Similarly, when a number is converted from any base to the decimal base then we write the number in that base in the expanded form and the result is the number in decimal form.

1. Example: Convert (1101)2 to decimal base
Sol. (1101)2 = 1 × 23 + 1 x 22 + 0 × 21 + 1 × 20 = 8 + 4 + 1 = 13
So (1101)2 = (13)10

2. Example: Convert the octal no 3456 in to decimal number.
Sol. 3456 = 6 + 5 × 8 + 4 × 82 + 3 × 83
= 6 + 40 + 256 + 1536
= (1838)10

3. Example: Convert (1838)10 to octal.
Sol.

Number Systems - Finding Remainders, Number Theory, Quantitative Aptitude Quant Notes | EduRev
= (3456)8

4. Example: What is the product of highest 3 digit number & highest 2 digit number of base 3 system?
(1) (21000)

(2) (22200)

(3) (21222)

(4) (21201)

(5) None
Sol. The highest 3 digit & 2 digit numbers are 222 & 22
222 = 2 + 2 × 3 + 2 × 32 = 26
22 = 2 + 2 × 3 = 8
∴ Product = 26 × 8 = 208
Convert back to base
(21201)3

Number Systems - Finding Remainders, Number Theory, Quantitative Aptitude Quant Notes | EduRev

5. Example: Find the fifth root of (15AA51)19
Sol. (15AA51)19 = 1.195+ 5.194+ 10.193+ 10.192+ 5.191+ 1.190

= (19+1)5 = 20 (Using binomial theorem)

Therefore, the fifth root is 20

6. Example: In a number system the product of 44 and 11 is 1034. The number 3111 of this system, when converted to the decimal number system, becomes?

(1) 406

(2) 1086 

(3) 213 

(4) 691

(5) None of the above

Sol.
Let the base be n

(4n+4)(n+1)= n3+3n+4

⇒ n3-4n2-5n=0

⇒ n(n-5)(n+1)=0
⇒ n=5

⇒(3111)= (406)10

IMPORTANT POINTS

The sum of consecutive five whole numbers is always divisible by 5.   

The square of any odd number when divided by 8 will leave 1 as the remainder

The product of any three consecutive natural numbers is divisible by 8.

The unit digit of the product of any nine consecutive numbers is always zero.

For any natural number n, 10n-7 is divisible by 3.

Any three-digit number having all the digits same will always be divisible by 37.

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