# Base System Concepts - ADURO

• The number system in which we work is called Decimal system. This is because there are 10 digits in decimal system(0-9).
Other base system are also there.
1- Binary-( 0,1) Two digits
2- octal-( 0-7) Eight digits
Here a= 10
B=11
C=12
D=13
E=14
F=15

The highest digit of nth base system is (n-1)
Eg- in binary highest digit is 1.

Eg- which number is possible in base7.
a) 35673
b) 64454
c) 372932
d) 72342

SOL:- b
Highest digit in base 7 can be 6 only.

Representation of a number in any base:

Let see representation of 543 in base 10 i.e. usual way
543 = 5 * 100 + 4 * 10 + 3 = 5 * 10^2 + 4 * 10^1 + 3 * 10^0
so in any base system x, the number is represented as
(abcd…….. n digit) in base x = a * x^(n-1) + b * x^(n-2) + …

Conversion from decimal to any base system:-

Convert 20 of base 10 in to base 8.

Procedure:-

1. Find the highest power of base in the number ( highest power of base 8 in 20 is 1). 8^1=8 and 8^2= 64( greater than 20 so highest power is 1).
2. If the highest power contained is n then there are total (n+1) digits in number.
( highest power is 1 so 2 digit i.e. (_ _ ) This type of number)
3. Find how many times the highest power is contained in number.( 8 * 1 = 8, 8 * 2 = 16, 8 * 3 = 24 (greater than 20) , so it is contained 2 times). This will be value of coefficient which holds the highest power. ( 2 _)
4. check for the number for which we still have to account.( 8 * 2=16 has been taken in to account , Remaining= 20-16=4.)
5. Repeat step 2. ( now after 8^1, it will be 8^0=1, How many times 8^0 or 1 will be contained in 4 , 1 * 4 = 4, 4 times, so number in base 8 will be 24)

Convert 74 of base 10 in base 7

7^1 = 1, 7^2 = 49, 7^3 = 343, so highest power is 2, Number of digits=3.
For 7^2:-
49 * 1 = 49, 49 * 2 = 98, so coefficient is 1.
74 - 49 = 25
Now for 7^1:- 7 * 1=7,7 * 2=14, 7 * 3 = 21 , So coefficient of 7^1 is 3.
74 - 49 - 21 = 4
Now for 7^0= 1:- 1 * 4= 4, so coefficient of 7^0=4
74 in base 10= 134 in base 7.

convert 457 to base 16.

Remember in base 16 the highest number is 15.
Till 0-9 the digits are same as we used to write in decimal system but after 9
a- 10
b-11
c- 12
d- 13
e- 14
f- 15
and when you write the digits as a,b,c,d,e,f, these all are single digit
number in base 16.
16^1=16, 16^2=256 so highest power is 2 and total digits= 3
256 * 1 = 256, 256 * 2=512, coefficient of 16^2= 1
Remaining number= 457-256=201
Coefficient of 16^1:-
16 * 12=192,16 * 13= 208, coefficient= 12 = C
Remaining number= 201-192=9
Coefficient of 16^0 = 1 * 9 = 9
Number= 1C9

NOTE:- When (x^n) base 10 is represented in base x, It is written as 1 followed by n zeroes (1….. n zeroes)
(3^2) base 10= (100) base 3
(2^100) base 10 = (10000……(100 times) ) base 2.

Conversion from any base to decimal:-

Convert (15) of base 8 in to base 10

For the part 15:-
It is same as representing 15 in base 8
(15) base 8= 1 * 8^1 + 5 * 1 = (8 + 5)base 10= (13) base 10

Note - Size of a number is inversely proportional to its base.
Eg- (3^2) base 10= (100) base 3
In base 10, size is a single digit while in base 3, it is a 3 digit number.

(75) base 10= (2abc) in base x. Find x and the sum (a+b+c) in base x.

(75) base 10= (2abc) in base x. Find x and the sum (a+b+c) in base x.
Size increased, so x is less than 10.
Convert (2abc)of base x in to base 10
= 2 * x^3 + a^x^2 + b * x + c * x^0
= 2 * x^3 + a^x^2 + b * x+c * 1
This is equal to 75
75 = 2 * x^3 + a^x^2 + b * x + c * 1

Let x = 2
Highest power of 2 in 75 will be 6.(2^6=64)

x=3
Highet power=3 (3^3=27)
Coefficient:-
27 * 2=54 (so coefficient and highest power are coming equal so we will check if there is any other x or not)
So yes may be.

x = 4
Coefficient:-
Highest power= 3(4^3=64)
Coefficient:-
64 * 1=64 ( so no x till now is 3)
So when x cant be 4 it wont be greater than 4.
So now the only task is converting (75) from base 10 to base 3
I am writing directly, and you should also do this calculation in mind only.
(75)base 10=(2210) base 3
a - 2
b - 1
c - 0
a + b + c = 3
3 in base 3= 10

(895) base 10=(5ab) in base x, Find (a+b+x) in base 10.

Visualize highest power is 2 and coefficient 5.
So actually coefficient of highest power reduced so x is greater than 10.
895= 5 * x^2 + a * x + b

X= 11:-
Highest power= 2 (11^2= 121)
Coefficient = 121*7= 847 , so base is not 11.

X=12:-
Highest power= 2( 12^2=144)
Coefficient= 144*6= 864, so rejected.

X= 13:-
Highest power= 2 (13^2= 169)
Coefficient= 169*5= (170-1)*5= 850-5= 845,

So yes !! base is 13.
Now do the conversion part by your self.

Direct convert in base:- : It is used if different base let base a and base b can be written as a^n=b or b^n =a, Then we combine the n digits starting from right, and if in the right most combination there are not n digits then append the required number of zeroes.

What will be (11011011101101001)base2 to base 8.

2^3= 8
So combine 3 - 3 digits starting from right.
((append 0)11 011 011 101 101 001)
001= 0 * 2^2 + 0 * 2^1 + 1 * 1 = 1
101= 1 * 2^2 + 0 * 2^1 + 1 * 1 = 4 + 1= 5
101= 1 * 2^2 + 0 * 2^1 + 1 * 1= 4 + 1 = 5
011= 0 * 2^2 + 1 * 2^1 + 1 * 1 = 2 + 1 = 3
011= 0 * 2^2 + 1 * 2^1 + 1 * 1 = 2 + 1 = 3
011 = 0 * 2^2 + 1* 2^1 + 1 * 1 = 2 + 1 = 3
= (3 3 3 5 5 1) base 8
This can be used for if base are (2,16) , (3,9) or any other just cluster the n digits where a^n=b.

Note:- If a number written in base “n” has “ k” digits then the number lies between n^k and n^(k-1).
Eg:- two digits number in base 10 = 10^2 - 10 = 90
3 digits number in base 5 = 5^3 - 5^2 = 125 - 25 = 100

Find the range of (2abc) base 6.

It is a 4 digit number so it will definitely lie between 6^4 and 6^3
2abc = 2 * 6^3 + a * 6^2 + b * 6^1 + c * 1
so as the a,b,c will vary the value will also vary.
What here counting is not a good idea.
The number which will be just greater than 2abc will be (3000).
3000= 3 * 6^3
And the least possible number of the form 2abc is (2000)
2000= 2 * 6^3
So the range is 2 * (6^3) and 3 * (6^3), NO!!!!!
3 * (6^3) is not of the form (2abc) so the range will be 2 * (6^3) and 3 * (6^3)-1.

How many numbers have exactly 3 digits when expressed in either base 6 or base 12

3 digit number in base 6:-
Highest number= 6^3-1=215
Lowest number= 6^2=36
3 digit number in base 12:-
Highest number= 12^3-1= 1728-1=1727
Lowest number= 12^2= 144
So common number will be from 144 to 215.

abc is a 3 digit number in base 9. When the digits are reversed then number is in base 11 and both the numbers are equal. Find (a+b+c) in base 10.

(abc) base 9= (cba) base 11
a * 81 + b * 9 + c = c * 121 + b * 11 + a
a,b,c must be less than 9 and a,c can not be 0.
a * 81 + b * 9 + c = c * 121 + b * 11 + a
120c - 80a + 2b=0
60c - 40a + b = 0
c= 2, a=3, b=0 satisfies
You can also check
(302)9= (203)11
But c=4 and a=6 will also satisfy.
Also c= 6 and a=9, but a,b,c can not be 9, so ignore.

how many 2 digits number in base 9 gets reversed when expressed in base 11.

(ab)9= (ba)11
9a+b= 11b+a
8a=10b
a/b= 5/4
a and b are single digit so only choice is 5,4.
So 1 number.

Divisibility in base system: A number in base x when converted to base 10 will be divisible by (x-1) if the sum the digits is divisible by (x-1).

What will be the remainder if (757)base 8 after converting to base 10 is divided by base 7.

(757)base 8 = (7 * 8^2 + 5 * 8 + 7) base 10.
[(7 * (7+1)^2 + 5 * (7+1)+7) base 10 ] mod 7
(7 + 1) mod 7=1
(7 * 1 + 5 * 1 + 7) mod 7=19 mod 7= 5

What will be the value of a if remainder is 0 when (54432a34)8 is converted to base 10 and then divided by 7.

So you know remainder will be 0 only if sum is divisible by 7.
Sum= 5+4+4+3+2+a+3+4=25+a
For sum to be divisible by 7.
a= 3

Number of trailing zeroes in base n: So in decimal system we used to check for highest power of 10, so in any base system we will check for the highest power of n.

Number of zeroes in 6! In base 2
6/2 + 6/4 = 3 + 1 = 4

Number of zeroes in 6! In base 6.

6 = 3 * 2
So check by 3 and 2. But we know 3 will be less so check directly by 3.
6/3=2
So 2 zero.

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