Number PlayClass 8 Mathematics Important Points

Any number can be expressed in an expanded form based on place value. For example, a 3-digit number 'abc' can be written as $100a + 10b + c$. This form is the basis for understanding divisibility rules.

A number is divisible by 2 if its units digit is even (0, 2, 4, 6, 8). It is divisible by 5 if its units digit is 0 or 5. It is divisible by 10 if its units digit is 0.

A number is divisible by 4 if the number formed by its last two digits (tens and units) is divisible by 4. A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

A number is divisible by 3 if the sum of its digits is a multiple of 3. For example, 486 is divisible by 3 because $4+8+6=18$, and 18 is a multiple of 3.

A number is divisible by 9 if the sum of its digits is a multiple of 9. The remainder when a number is divided by 9 is the same as the remainder when the sum of its digits is divided by 9.

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places (from right to left) is either 0 or a multiple of 11. For 1331, $(1+3) - (3+1) = 0$, so it is divisible by 11.

To check divisibility by a composite number, test for divisibility by its co-prime factors. For example, to check for 6, test for both 2 and 3. To check for 12, test for 3 and 4.

If a number $a$ divides two numbers $M$ and $N$, then $a$ also divides their sum $(M+N)$ and their difference $(M-N)$. For example, since 8 divides 16 and 24, it also divides $16+24=40$ and $24-16=8$.

If a number $A$ is divisible by another number $k$, then all multiples of $A$ are also divisible by $k$. For example, since 14 is divisible by 7, all multiples of 14 (like 28, 42, 70) are also divisible by 7.

If a number $A$ is divisible by $k$, then $A$ is also divisible by all the factors of $k$. For example, if a number is divisible by 36, it must also be divisible by the factors of 36, which are 1, 2, 3, 4, 6, 9, 12, and 18.

A number that leaves a remainder $r$ when divided by a divisor $d$ can be expressed in the general form $dk+r$, where $k$ is a whole number. For example, numbers that leave a remainder of 2 when divided by 6 are of the form $6k+2$.

Parity refers to whether a number is even or odd. The rules are: $\text{even} \pm \text{even} = \text{even}$, $\text{odd} \pm \text{odd} = \text{even}$, and $\text{even} \pm \text{odd} = \text{odd}$.

Cryptarithms are puzzles where letters stand for digits. Solve them using logic, place value, and properties of arithmetic operations. For example, in $A1 + 1B = B0$, from the units column $1+B=10$, so $B=9$.