Divisibility by \(3\):
If the sum of the digits of the number are divisible by \(3\), then that number is divisible by \(3\).
Example:
Consider the number \(681\).
In this, the sum of the digits \(= 6 + 8 + 1 = 15\)
\(\frac{15}{3} = 5\)
Thus, \(681\) is divisible by \(3\).
Divisibility by \(6\):
If a number is divisible by \(2\) and \(3\), then that number is divisible by \(6\).
Example:
Consider the number \(768\).
Since the given number ends with an even number, then it is divisible by \(2\).
Sum of the digits \(= 7 + 6 + 8 = 21\), which is divisible by \(3\).
Since, \(768\) is divisible by both \(2\) and \(3\), then it is also divisible by \(6\).
Divisibility by \(9\):
A number is divisible by \(9\) if the sum of its digits is divisible by \(9\).
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Sometimes, if the sum is large, keep adding the digits of the sum until you get a number less than \(9\) or equal to \(9\), then check divisibility.
Example:
Consider the number \(987450633\).
Sum of the digits \(= 9 + 8 + 7 + 4 + 5 + 0 + 6 + 3 + 3 = 45\)
Again adding the result \(= 4 + 5 = 9\) which is divisible by \(9\)
Thus, the number \(987450633\) is divisible by \(9\).
Important!
If a number is divisible by \(9\), then it is divisible by \(3\). However, there are few multiples of \(3\) that are not multiples of \(9\). Eg: \(15\), \(21\), \(24\).
Divisibility by \(11\):
A number is divisible by \(11\) if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is either \(0\) or a multiple of \(11\).
Example:
Consider the number \(378301\).
Sum of digits in even places \(= 3 + 8 + 0 = 11\)
Sum of digits in odd places \(= 7 + 3 + 1 = 11\)
Difference \(= 11 - 11 = 0\)
Thus, the number \(378301\) is divisible by \(11\).
Divisibility by \(12\):
A number is divisible by \(12\) if and only if it is divisible by \(3\) and \(4\).
Example:
Consider the number \(564\).
Sum of the digits \(= 5 + 6 + 4 = 15\) which is divisible by \(3\).
Consider the last two digits \(64\), which is divisible by \(4\).
Therefore, the number \(564\) is divisible by \(12\).
Divisibility by \(24\):
A number is divisible by \(24\) if and only if it is divisible by \(3\) and \(8\).
Example:
Consider the number \(3144\).
Sum of the digits \(= 3 + 1 + 4 + 4 = 12\) which is divisible by \(3\).
Consider the last three digits \(144\), which is divisible by \(8\).
Therefore, the number \(3144\) is divisible by \(24\).