Base System  Vikas Saini

We use base 10 for mathematical operation. So we use 10 digit (09). After 9, we use for next number 10. Then 11,12,13.....19. Then again 20,21....and so on.
So we use digits as per that number system.
In base 2  2 digits (0,1)
base 3  3 digits (0,1,2)
base 4  4 digits (0,1,2,3)
base 10  10 digits (0,1,2,3,4,5,6,7,8,9)
Base 13  13 digits (0,1,2,3,4,5,6,7,8,9,A,B,C,D)
base 16  16 digits(0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F)
Cream of the piece 
1. In any base system n, number of digits is equal to n.
2. The max digit in any number system 'n' is 'n1' .
Conversion in any base 'n' from base '10'
Convert 77(base 10) in base 6.
77 = (abc)_6
77 = a x 6^2 + b x 6 + c.
if a = 1,
then b = 5, c = 5.
(155)_6 = 71.(not equal to 77).
if a=2, b = 0, c = 5.
(205)_6 = (77)_10.
Convert (158 )_10 in base 16.
(158 )_10 = (abc)_16.
a x 16^2 + b x 16 + c.
here a = 0, b = 9, c = E.
(9E)_16 = (158 )_10.
Convert (1078 )_10 in base 7.
(1078 )_10 = (abcd)_7
a x 7^3 + b x 7^2 + c x 7 + d.
a = 3, b = 1, c = 0, d = 0.
(3100)_7 = 1078.
Conversion from base 'n' to base '10'
(abcd)_n = a x n^3 + b x n^2 + c x n^3 + d.
Convert (66)_7 in base 10.
6 x 7 + 6 = 48.
(66)_7 = (48 )_10.
Convert (599)_11 in base 10.
5 x 11^2 + 9 x 11 + 9
= 605 + 99 +9
= 713.
Convert (1274)_9 in base 10.
1 x 9^3 + 2 x 9^2 + 7 x 9 + 4
= 729 + 162 + 63 + 4
= 958.
(1274)_9 = (958 )_10.
Now some questions from Base system.
Q. 677 has exactly 5 digits when converted into base 'n' from the decimal system. What is the minimum possible value of n.
Solution :
Let's start from base 3.
smallest 5 digit number in base 3 = (10000)_3 = 3^4.
Largest 5 digit number in base 3 = (22222)_3 = 2(11111)_3 = 3^5  1.
in base 3 not possible.
Now let's take base 4.
smallest number = (10000)_4 = 4^4 = 256.
Largest number = (33333)_4 = 4^5  1 = 1023.
256 < 677 < 1023.
Hence base 4.
Q. How many 3 digit numbers are there in base 9 system.
Solution :
Hundred place = 1 to 8 = 8 digit.
Ten's place = 0 to 8 = 9 digit.
Unit's place = 0 to 8 = 9 digit.
Total = 8 x 9 x 9 = 648 digits.
Q. How many decimal numbers are three digit numbers in base 7 and four digit numbers in base 6.
Solution :
Smallest 3 digit number in base 7 = (100)_7 = 7^2 = (49)_10.
Largest 3 digit number in base 7 = (666)_7 = 7^3  1 = (342)_10.
[49,342] numbers are 3 digits numbers in base 7.
Smallest 4 digit number in base 6 = 6^3 = 216.
Largest 4 digit number in base 6 = 6^4  1 = 1295.
[216,1295] numbers are 4 digit numbers in base 6.
[216,342] common numbers.
Total decimal numbers = 342  216 + 1 = 127.
Q. (N)_10 = (aaa)_3.
here a is a single digit number, then N is a multiple of.
(a) 9 (b)3a
(c)13 (d)26.Solution :
(aaa)_3 = a (111)_3
a (3^2 + 3 + 1)
(13a)_10 = (N)_10.
Hence 13.
Q. How many 3 digits are there in decimal system which have exactly 3 digits when expressed in both base 7 & base 13.
Solve:
smallest 3 digit number in base 7 = (100)_7 = 7^2 = 49.
Largest 3 digit number in base 7 = (666)_7 = 7^3  1 = 342.
[100,342] are the number which are exactly 3 digit number in base 7.
smallest 3 digit number in base 13 = (100)_13 = 13^2 = 169.
Largest 3 digit numbers in base 13 = (CCC)_13 = 12^3  1 = 1727.
[169,999] are the three digit numbers which are exactly 3 digit numbers in base 13.
[ 169,342] is common in both.
Total = 342  169 + 1 = 174 numbers.
Q. (206)_x + (216)_x = (424)_x .
what is value of x ?
Solution :
2x^2 + 6 + 2x^2 + x + 6 = 4x^2 + 2x +4
= > x + 12 = 2x + 4.
= > x = 8.
Q. (xy)_8 + (yx)_8 = (abc)_10.
How many values are possible of abc.
Solution :
8x + y + 8y + x = abc.
9x + 9y = abc.
9(x+y) = abc.
x = 5, y = 7 then abc = 108.
x = 6, y =7 then abc = 117.
but abc are distinct digits.
so only 1 value 108.
Q. A two digit number in base 7 is equal to the two digit number formed by reversing it digits but in base 13.
How many such numbers are possible ?
Solution :
(xy)_7 = (yx)_13
7x+y = 13y+x
6x = 12y
x = 2y.
x = 1,2,3.
y = 2,4,6.
three numbers 12,24 and 36.
Divisibility rules in various Base system
In any base system 'n' a number is divisible by 'n1' depends upon sum of digits of number. As we use base 10; if we have to check any number is divisible by 9,we find sum of digits of number and divide by 9.
suppose let's check 297 is divisible by 9 or not.
2 + 9 + 7 = 18.
we get digit sum = 18.
18 mod 9 = 0.
297 is divisible by 9.
Find the remainder when (8484)_12 is divided by 11.
8 + 4 + 8 + 4 = 24.
24 mod 11 = 2.
Find remainder when (99987)_11 divided by 10.
9 + 9 + 9 + 8 + 7 = 42.
42 mod 10 = 2.
In any base n, any number divided by (n + 1) leaves the remainder as same as the difference between sum of the digits of number in odd places and that in even places is divided by (n + 1). Let's check 13486 is divisible by 11 or not.
(1 + 4 + 6)  (3 + 8 ) = 0.
divisible by 11.
Find the remainder (8484)_12 is divided by 13.
(4 + 4)  (8 + 8 ) = 8.
8+13 = 5.
remainder = 5.
find the remainder (76456)_8 divided by 9.
(7 + 4 + 6)  ( 6 + 5) = 6.
remainder = 6.
In any base n, a number divided by a factor by factor of n leaves same remainder as the last digit of number is divided by factor of n. suppose in base 10, any no is divisible by 2 & 5.
Then last digit of both number both should be divisible by 2 & 5.
To check any no is divided by 4, we check last 2 digits are divisible by 4 or not. ( 4 = 2^2).
To check any no is divided by 8, we check last 3 digits are divisible by 8 or not.
Q. A 99 digit number is formed by writing first 54 natural numbers in front of each other as 123 ... 54.
Find the remainder when this number is divided by 8. ( CAT 1998 ).
(a) 4 (b) 7
(c) 2 (d) 0
Solution :
We know 2 is factor of 10.
8 = 2^3.
Hence 8 must be factor of 10^3.
We need to divide last 3 digits only.
354 mod 8 = 2.

Sir, please explain the last question again.

For the divisibility by 8 , we check only for the last three digits of the number. If the last three digits of a number is divisible by 8, then the number will be divisible otherwise not. For the remainder , the remainder left by the division of last three digits by 8 will be the remainder when the whole number is divided by 8.
So, here the number is 1234567891011121314…….525354.
Last three digits of the number = 354
Remainder when the number is divided by 8 = Rem(354/8) = 2

@vikas_saini Just curious, why the last question is added in a Base System chapter. Seems more like a Divisibility rule concept than base system.
Your articles are very good. Thanks for all the effort

@harris Thank you very much sir..it was old cat question, so I added it.

@vikas_saini Good. Just asked