Loading....
Coupon Accepted Successfully!

 

Divisibility Rules For Systems Other Than Decimal System

I would like to emphasize that different number systems are just different ways to write numbers. Thus the divisibility of one number by another does not depend on the particular system in which they are written.
 
At the same time, in each system there are some tricks to determine divisibility by certain specific numbers. These are the divisibility tests.
 
Let us investigate the tests for divisibility by 3 and 9. We will try to generalize these tests for any number base system.

Is 1234565642317 divisible by 6?
 
We know the divisibility rule for 9 – Sum of digits of the number should be divisible by 9.
Sum of the digits of this number is 42.
 
Now, we can answer this question easily, since the sum of the digits (which is 4210) is divisible by 6, the number itself is also divisible by 6.
 
In general, the sum of the digits of a number written in the base n system is divisible by (n – 1) if and only if the number itself is divisible by (n – 1).
 
So, divisibility rule for 4 in a base system of 5 in that the sum of the digits of the number should be divisible by 4. For example, 315 is divisible by 4.
 
Similarly, if we have to find out the divisibility rule of 12 in the base of 11, it will be nothing but same as the divisibility rule of 11 in the base of 10. Generalizing this whole concept, divisibility rule of any natural number N in the base of (N – 1) will be same as divisibility rule of 11 on base 10.
 
Before you proceed, you should have a clear understanding of how to find the divisibility rule for any prime number in base 10.
 
The only difference is that in case of base 10, we find the multiples of prime numbers in the format 10 n ± 1, and in case of base B, we find the multiples of prime numbers in the format B n ± 1.

Let us find out the divisibility rule of (5)8.
Multiples of 5 in base 10
Description: 5673.png 

Obviously, 17 is the first number in the format of 8 n ± 1.
17 = 8 × 2  1[Yes, it is correct. 178 and RHS is in base 10]
So, divisibility rule of 5 on base 8 = A + 2B, where A and B have the same meaning as that in divisibility rules of 10.

We can check this by the following method:
Description: 5667.png 

Every 5th number in this series will be divisible by 5. So, 5 is the first number divisible by 8. The next few numbers are 12, 17, 24, 31, 36.
 
We can use base to generate good concepts in remainders also.
 
Example-1
What is the remainder when (1234)6 is divided by 5?
Solution
Divisibility rule of (5)6 will be similar to that of the divisibility rule of (9)10. So, if the sum of digits of the number in base 6 is divisible by 5, then it will be divisible by 5.
 
Adding the digits of (1234)6 = 10, which is divisible by 5, the remainder obtained when (1234)6 is divided by 5 is 0.
 

Example-2
What is the remainder when (12341)6 is divided by 5?
Solution
The sum of the digits of (12341)6 = 11, so the remainder obtained will be 1.
 




Test Your Skills Now!
Take a Quiz now
Reviewer Name