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 \(9\).
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 \(24\).
Digital Roots
The digital root in math is the single-digit value obtained by continuously summing the digits of a number until only one digit remains.
How Digital Roots Work?
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Start by adding the digits of a number.
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If the result has more than one digit, repeat the process with the new sum.
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Continue until a single-digit number is reached.
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For example, for \(67392\): \(6 + 7 + 3 + 9 + 2 = 27\) then, \(2 + 7 = 9\). So, the digital root is \(9\).
Applications
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Used to check arithmetic calculations, especially for addition, subtraction, and multiplication by confirming digital roots match expectations.
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Helpful in divisibility tests, particularly for \(9\) (and sometimes for \(3\)).
Important!
The digital root of multiples of \(9\) is \(9\).