Divisibility by \(2\):
Divisibility by \(2\): If the number ends at \(2\), \(4\), \(6\), \(8\) or \(0\), it is divisible by \(2\).
What is Parity?
Parity tells us if a number is even or odd:
Parity tells us if a number is even or odd:
- An even number leaves no remainder when divided by \(2\). It is called as even parity (eg: \(6 \div 2 = 3\) with remainder \(0\)).
- An odd number leaves a remainder of \(1\) when divided by \(2\). It is called as odd parity (eg: \(7 \div 2 = 3\) with remainder \(1\)).
Addition and Subtraction Parity:
odd \(\pm\) odd \(=\) even
even \(\pm\) even \(=\) even
odd \(\pm\) even \(=\) odd
Product of Parity:
odd \(\times\) even \(=\) even
odd \(\times\) odd \(=\) odd
Exponent Parity:
odd \(^3 =\) odd
odd \(\pm\) odd \(=\) even
even \(\pm\) even \(=\) even
odd \(\pm\) even \(=\) odd
Product of Parity:
odd \(\times\) even \(=\) even
odd \(\times\) odd \(=\) odd
Exponent Parity:
odd \(^3 =\) odd
Divisibility by \(4\):
If a last two digits of any number are divisible by \(4\), then that number is divisible by \(4\).
Example:
Consider the number \(248\).
In this, the last two digits are \(48\).
\(\frac{48}{4} = 12\)
Thus, \(248\) is divisible by \(4\).
Divisibility by \(5\):
If a digit in the ones place of a number is \(5\) or \(0\), then it is divisible by \(5\).
Example:
Consider the number \(3270\).
In this, the last digit is \(0\).
Thus, \(3270\) is divisible by \(5\).
Divisibility by \(8\):
A number is divisible by \(8\) if the number formed by its last three digits is divisible by \(8\).
Example:
Consider the number \(35416\).
Consider the last three digits \(416\). Here, \(\frac{416}{8} = 52\).
Since \(416\) is divisible by \(8\), the number \(35416\) is divisible by \(8\).
Divisibility by \(10\):
A number is divisible by \(10\), if it ends with \(0\).
Example:
Consider the number \(3320\).
In this, the last digit is \(0\).
Thus, \(3320\) is divisible by \(10\).
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 \(11\).