Mathematics: Class 8 (Part 1) — Chapter 5: Number Play
5.1 Is This a Multiple Of?
Sum of Consecutive Numbers & Parity
Consecutive Sums: Any odd number can be written as the sum of two consecutive natural numbers (e.g., $7 = 3 + 4$).
Parity of Expressions: For any four numbers $a, b, c, d$, all combinations of $a \pm b \pm c \pm d$ share the same parity (either all are even or all are odd).
Parity Rules:
$\text{Odd} \pm \text{Odd} = \text{Even}$
$\text{Even} \pm \text{Even} = \text{Even}$
$\text{Odd} \pm \text{Even} = \text{Odd}$
Pairs to Make Fours (Divisibility by 4)
Even numbers can be classified into two types based on their remainder when divided by 4:
Multiples of 4: Expressed as $4p$ (remainder 0).
Non-multiples of 4: Expressed as $4p + 2$ (remainder 2).
Sum of Two Even Numbers:
Divisible by 4 if both are multiples of 4: $4p + 4q = 4(p+q)$.
Divisible by 4 if both are non-multiples of 4: $(4p+2) + (4q+2) = 4(p+q+1)$.
Not divisible by 4 if one is a multiple of 4 and the other is not: $4p + (4q+2) = 4(p+q) + 2$.
5.2 Checking Divisibility Quickly
Standard Divisibility Rules
Divisibility by 10: The units digit must be $0$.
Divisibility by 5: The units digit must be $0$ or $5$.
Divisibility by 2: The units digit must be even ($0, 2, 4, 6, 8$).
Divisibility by 9: The sum of the digits must be divisible by $9$.
Divisibility by 3: The sum of the digits must be divisible by $3$.
Divisibility by 11: The difference between the sum of digits at odd places and the sum of digits at even places must be $0$ or a multiple of $11$.
Digital Roots
Digital Root: The single-digit value obtained by repeatedly adding the digits of a number (e.g., for $489710$: $4+8+9+7+1+0 = 29 \rightarrow 2+9 = 11 \rightarrow 1+1 = 2$).
The digital root of any multiple of $9$ is always $9$.
5.3 Digits in Disguise (Cryptarithms)
Rules & Solving Cryptarithms
Cryptarithm: A mathematical puzzle where letters stand for unique digits ($0-9$).
The leading digit of any number cannot be $0$.
Example Puzzle: $AB + 37 = 6A$
Since $B + 7$ ends in $A$, and $A + 3 (+ \text{carry}) = 6$, we find $A = 2$ and $B = 5$ ($25 + 37 = 62$).
Summary & Key Properties
If $a$ divides $b$, then all multiples of $a$ are divisible by $b$.
If $a$ is divisible by $b$, then $a$ is divisible by all factors of $b$.
If $a$ divides $m$ and $n$, then $a$ divides $m \pm n$.
If $a$ is divisible by $b$ and $c$ (where $b, c$ are co-prime), then $a$ is divisible by $b \times c$ (or the LCM of $b$ and $c$).
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