A divisibility test is a method to determine whether one number is divisible by another without actually performing the division. It's a quick way to check if a number can be evenly divided by another number without leaving a remainder.
Formula for Divisibility
The general formula for divisibility is:
\[a = b \times q + r\]
Where:
\(a\) is the dividend (number being divided)
\(b\) is the divisor
\(q\) is the quotient
\(r\) is the remainder
A number \(a\) is divisible by \(b\) if and only if \(r = 0\).
Common Divisibility Rules
Divisible by 2: The last digit is even (0, 2, 4, 6, or 8).
Divisible by 3: The sum of all digits is divisible by 3.
Divisible by 4: The number formed by the last two digits is divisible by 4.
Divisible by 5: The last digit is 0 or 5.
Divisible by 6: The number is even and the sum of its digits is divisible by 3.
Divisible by 9: The sum of all digits is divisible by 9.
Divisible by 10: The last digit is 0.
Example with Visual Representation
Let's test if 144 is divisible by 12:
\[144 \div 12 = 12\]
Remainder: \[144 - (12 \times 12) = 0\]
This diagram visually represents that 144 is perfectly divisible by 12, as the entire rectangle is filled, indicating no remainder.
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