# Base System - Anubhav Sehgal, NMIMS Mumbai

• General principle
The number of digits used in a particular system of writing numbers is known as the base. Our well known decimal system (or base 10) has 10 digits (0 to 9). In general, a base-n system uses digits 0 to (n – 1). Like our count in decimal system runs in cycles of 10, another base n runs in cycles of n and moves to the next significant digit to the left when its digits get exhausted for the count. Some basic principles

1. A number in base system n, say abc, is written as c + b * n + a * n^2. This would be familiar to writing a number, say 145, in decimal system as 5 + 4 * 10 + 1 * 100. Same principle applies for a general base n.
2. A base system N uses digits 0 to (N – 1) only for its representation.
3. A number is base N is divisible by N-1, when the sum of digits in base N is divisible by N-1
4. When digits of a number N1 in base N are rearranged to form a number N2, then N2-N1 is always divisible by N-1.
5. If a number in base N has even number of digits and that number is a palindrome, then the number is divisible by N+1

A number 2342a121 is in base 8 and it is divisible by 7. Find the value of a.

Using principle 3, (2 + 3 + 4 + 2 + a + 1 + 2 + 1) is divisible by 7.
(15 + a) is divisible by 7
a= 6

The value of (222) in base ‘x’ when converted to base 10 is ‘P’. The value of (222) in base ‘y’ when converted to base 10 is Q. If (P – Q) in base 10 = 28, then what is the value of (Q – x) in base 10?

Use principle 1, P = 2 + 2x + 2x^2
Q = 2 + 2y + 2y^2
(P – Q) = 28
2(x^2 – y^2) + 2(x – y) = 28
(x – y)(x + y + 1) = 14
= 1 * 14
= 2 * 7
[Note: Second bracket is always greater than the first hence we need not check for reverse combinations of 14 * 1 and 7 * 2.]
(x – y) = 1 and (x + y + 1) = 14
(x – y) = 1 and (x + y) = 13
x = (13 + 1)/2 = 7, y = (13 – 1)/2 = 6
Q = 2 + 2(6) + 2(36) = 86
(Q – x) = 79

(x – y) = 2 and (x + y + 1) = 7
(x – y) = 2 and (x + y) = 6
x = (6 + 2)/2 = 4, y = (6 – 2)/2 = 2

Not possible as if base y is 2 then it cannot have a number 222 existing in it since a base n uses only digits 0 to (n – 1). Hence our answer will be distinct and equal to 79. If more than one valid values appear, you get your answer as cannot be determined.

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