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Find Divisibility Rules for the Numbers from 2 to 13

Derive rules to test whether a number is divisible by N, where N ranges from 2 to 13. E.g. A number is divisible by 3 if the sum of the digits is divisible by 3. Show that a palindromic number which has an even number of digits is always divisible by 11.

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Divisibility Rules

The following rules all derive from the following property:

Further
Let's say we have a decimal number where
1) Is a divisible by 2? Well that's easy, if it's an even number, it's divisible by 2, if it's odd, it's not. Well to start as we mean to go on, a is divisible by 2 if

So a is divisible by 2 if the least significant digit is divisible by 2. I.e. a is even.

2) Is a divisible by 3? It is if:

So a is divisible by 3 if the sum of the digits is divisible by 3

3) Is a divisible by 4? It is if:
So a is divisible by 4 if is divisible by 4 (or which is just the number formed by the 2 least significant digits).

4) Is a divisible by 5? It is if:

So a is divisible by 5 if the least significant digit is divisible by 5, i.e. if the least significant digit is 0 or 5.

5) Is a divisible by 6? Well if it's divisible by 2 and 3, it's divisible by 6. So simply apply the rules for divisibility by 2 and 3.

6) Is a divisible by 7? It is ...

Solution Summary

Divisibility Rules for the Numbers from 2 to 20 are found. The solution is detailed and well presented.

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