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 \(9\):
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 \(9\).
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 \(11\).
Important!
If the number is not divisible by \(11\), then the difference obtained is the remainder.