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📖 Mathematics — Part 1

Std 8
Curriculum Book Guide
1 2 3 4 5 6 7
Chapters
1 A Square and a Cube 2 Power Play (Exponents) 3 A Story of Numbers 4 Quadrilaterals 5 Number Play 6 We Distribute Yet Things Multiply 7 Proportional Reasoning - I
5
Chapter 5
Skill: 30%

Number Play

Number Play

Chapter at a Glance

This chapter explores prime factors, multiples, divisibility rules, and remainder arithmetic using algebraic proofs, visual models, and puzzles. It begins by analyzing the parities of expressions formed by adding and subtracting consecutive numbers. The chapter details quick divisibility shortcuts (specifically for $3, 9,$ and $11$) and explains why they work by breaking numbers down into their place-value components. Additionally, it introduces the concept of digital roots, explores algebraic remainder representation (congruence concepts), and walks through techniques for solving cryptarithms (digit-letter substitution puzzles).

Key Definitions & Terminology

  • Parity: The classification of an integer as either even or odd.
  • Even Number: Any integer that is divisible by $2$ without a remainder, including negative even numbers (e.g., $2n$, where $n \in \mathbb{Z}$).
  • Odd Number: Any integer that leaves a remainder of $1$ when divided by $2$ (e.g., $2n + 1$).
  • Multiples: Numbers obtained by multiplying a given integer by other integers.
  • Factors: Integers that divide a given number exactly, leaving no remainder.
  • Digital Root: The single-digit value obtained by repeatedly summing the digits of a number until only one digit remains.
  • Remainder: The amount "left over" after performing an integer division.
  • Cryptarithm: A mathematical puzzle where digits are replaced by letters of the alphabet, and the goal is to determine the digit associated with each letter.
  • Modular Parity Rule: The classification of even numbers into two categories based on their division by $4$: those that are multiples of $4$ (remainder $0$) and those that are not (remainder $2$).

Formulas, Rules & Properties

  • Laws of Parity Operations:
  • $$\text{even} \pm \text{even} = \text{even}$$
  • $$\text{odd} \pm \text{odd} = \text{even}$$
  • $$\text{odd} \pm \text{even} = \text{odd}$$
  • $$\text{even} \times \text{any integer} = \text{even}$$
  • $$\text{odd} \times \text{odd} = \text{odd}$$
  • Sum Parity Invariance (Sign Swapping):
  • For any set of integers $a, b, c, d$, any expression of the form $a \pm b \pm c \pm d$ has the same parity. Swapping a sign (from $+$ to $-$ or vice versa) shifts the total value by an even number ($2b$), leaving the parity unchanged.
  • Remainder Representation Formula:
  • Any number $N$ that leaves a remainder $r$ when divided by a divisor $d$ can be written as:
    $$N = kd + r \quad \text{or} \quad N = kd - (d - r)$$
    where $k$ is an integer. For example, numbers leaving a remainder of $3$ when divided by $5$ are represented as $5k + 3$ or $5k - 2$.
  • Divisibility Rules:
  • Divisibility by 2, 5, 10: Depends solely on the units digit ($a$). A number is divisible by $2$ if $a \in {0, 2, 4, 6, 8}$, by $5$ if $a \in {0, 5}$, and by $10$ if $a = 0$.
  • Divisibility by 4: A number is divisible by $4$ if its last two digits form a multiple of $4$.
  • Divisibility by 8: A number is divisible by $8$ if its last three digits form a multiple of $8$.
  • Divisibility by 3 and 9: A number is divisible by $3$ (or $9$) if and only if the sum of its digits is divisible by $3$ (or $9$).
  • Divisibility by 11 (Alternating Sum): A number is divisible by $11$ if the difference between the sum of the digits in odd-numbered places and the sum of the digits in even-numbered places is $0$ or a multiple of $11$.
  • Composite Divisibility Rule:
  • If a number is divisible by two coprime numbers $x$ and $y$, it is divisible by their product $xy$. For example, a number is divisible by $24$ if it is divisible by both $3$ and $8$ (since $\text{GCD}(3, 8) = 1$). Checking divisibility by $4$ and $6$ is insufficient because they are not coprime ($\text{LCM}(4, 6) = 12$).

Core Concepts & Topics

  • Algebraic Explanations for Divisibility:
  • Divisibility by 10: Any number $\dots dcba$ can be written as $\dots + 1000d + 100c + 10b + a$. Since all terms from $10b$ upward contain a factor of $10$, the entire number is divisible by $10$ if and only if $a$ is divisible by $10$ (which means $a = 0$).
  • Divisibility by 9: Expanding place values as $(9+1), (99+1), (999+1)$ yields:
    $$7309 = 7(999+1) + 3(99+1) + 0(9+1) + 9(1) = 9(777 \times 9\text{-terms}) + (7 + 3 + 0 + 9)$$
    This shows the number is equal to a multiple of $9$ plus the sum of its digits.
  • Parity of Division Remains:
  • Summing two even numbers that are not multiples of $4$ (represented as $4p+2$ and $4q+2$) always yields a multiple of $4$:
    $$(4p+2) + (4q+2) = 4p+4q+4 = 4(p+q+1)$$
  • Digital Roots Behavior:
  • The digital root of any multiple of $9$ is $9$.
  • The digital root of a number represents the remainder when that number is divided by $9$ (if the digital root is less than $9$).

Worked Examples

  • Finding Consecutive Sum (Figure it Out Q1):
  • Problem: The sum of four consecutive numbers is $34$. Find the numbers.
  • Solution: Let the numbers be $x, x+1, x+2, x+3$.
    $$x + (x+1) + (x+2) + (x+3) = 34 \rightarrow 4x + 6 = 34 \rightarrow 4x = 28 \rightarrow x = 7$$
    • Answer: The numbers are $7, 8, 9,$ and $10$.
  • Smallest Multiple of 9 with No Odd Digits:
  • Problem: Find the smallest multiple of $9$ consisting only of even digits ($0, 2, 4, 6, 8$).
  • Solution:
    • The sum of the digits must be a multiple of $9$.
    • The sum of even digits must be even, so the smallest possible digit sum is $18$.
    • To minimize the number, we want the fewest digits. Since the max even digit is $8$, $2$ digits cannot sum to $18$ ($8+8=16$). Thus, we need at least $3$ digits.
    • The digits must sum to $18$. To make the number as small as possible, we place the smallest possible digit in the hundreds place: $2$. The remaining two digits must sum to $16$, which can only be $8$ and $8$.
    • Answer: $288$.
  • Chinese/Greek Tally Riddle (Figure it Out Q5):
  • Problem: "When I group my pebbles in 3's, remainder is 1. When I pair them, remainder is 1. Group them by 5, remainder is 1. Grouping by 7, remainder is 0. Total is less than 100. How many pebbles?"
  • Solution: Let the number of pebbles be $P < 100$.
    • $P$ leaves a remainder of $1$ when divided by $3, 2,$ and $5$.
    • Since $3, 2, 5$ are prime, $\text{LCM}(3, 2, 5) = 30$.
    • Thus, $P$ can be written in the form $30k + 1$.
    • For $P < 100$, the possible values are:
    • For $k = 1 \rightarrow P = 31$ (not divisible by 7).
    • For $k = 2 \rightarrow P = 61$ (not divisible by 7).
    • For $k = 3 \rightarrow P = 91$ (divisible by 7: $91 \div 7 = 13$).
    • Answer: $91$ pebbles.
  • Snehal's Divisibility Claim (Figure it Out Q2):
  • Problem: "Number $a$ leaves remainder $8$ when divided by $12$. Number $b$ is $4$ short of a multiple of $12$. Their sum is always a multiple of $8$." Is this true?
  • Solution:
    • Let $a = 12n + 8$ and $b = 12m - 4$.
    • $a + b = (12n + 8) + (12m - 4) = 12(n+m) + 4 = 12k + 4$.
    • Since $12k+4$ leaves a remainder of $4$ when divided by $8$ (if $k$ is even) or $0$ (if $k$ is odd), it is not always divisible by $8$. For example, if $k = 2$, $12(2)+4 = 28$ (not a multiple of $8$).
    • Answer: The claim is false.
  • Multiple of 18 Digit Pairs (Figure it Out Q5, Page 166):
  • Problem: If $48a23b$ is a multiple of $18$, find all possible pairs for digits $a$ and $b$.
  • Solution: A number is a multiple of $18$ if it is divisible by both $2$ and $9$.
    • For divisibility by $2$, the units digit $b$ must be even: $b \in {0, 2, 4, 6, 8}$.
    • For divisibility by $9$, the sum of the digits must be a multiple of $9$:
      $$4 + 8 + a + 2 + 3 + b = 17 + a + b = \text{multiple of 9}$$
    • We check each case for $b$:
    • If $b=0 \rightarrow 17+a+0 = 18 \rightarrow a=1$. Pair: $(1, 0)$.
    • If $b=2 \rightarrow 17+a+2 = 27 \rightarrow a=8$. Pair: $(8, 2)$.
    • If $b=4 \rightarrow 17+a+4 = 27 \rightarrow a=6$. Pair: $(6, 4)$.
    • If $b=6 \rightarrow 17+a+6 = 27 \rightarrow a=4$. Pair: $(4, 6)$.
    • If $b=8 \rightarrow 17+a+8 = 27 \rightarrow a=2$. Pair: $(2, 8)$.
    • Answer: The possible pairs are $(1, 0), (8, 2), (6, 4), (4, 6),$ and $(2, 8)$.
  • Divisibility by 44 Digit Pairs (Figure it Out Q6):
  • Problem: If $3p7q8$ is divisible by $44$, find all possible pairs for $p$ and $q$.
  • Solution: Divisible by $44$ means divisible by both $4$ and $11$.
    • For divisibility by $4$, the last two digits $q8$ must be divisible by $4$: $q8 \in {08, 28, 48, 68, 88} \rightarrow q \in {0, 2, 4, 6, 8}$.
    • For divisibility by $11$, the alternating sum must be a multiple of $11$:
      $$\text{Odd place sum} = 8 + 7 + 3 = 18$$
      $$\text{Even place sum} = p + q$$
      $$\text{Difference} = 18 - (p + q) = 0 \text{ or } 11$$
    • Case 1: $18 - (p+q) = 0 \rightarrow p+q = 18$. Since $p, q$ are digits and $q$ is even, if $q=8 \rightarrow p=10$ (not a digit). So this case is impossible.
    • Case 2: $18 - (p+q) = 11 \rightarrow p+q = 7$.
      • If $q=0 \rightarrow p=7$. Pair: $(7, 0)$.
      • If $q=2 \rightarrow p=5$. Pair: $(5, 2)$.
      • If $q=4 \rightarrow p=3$. Pair: $(3, 4)$.
      • If $q=6 \rightarrow p=1$. Pair: $(1, 6)$.
      • If $q=8 \rightarrow p=-1$ (impossible).
    • Answer: The possible pairs $(p, q)$ are $(7, 0), (5, 2), (3, 4),$ and $(1, 6)$.
  • Solving Multiplication Cryptarithm (Figure it Out Q15):
  • Problem (i): Solve $EF \times E = GGG$.
    • Solution:
    • $EF$ is a 2-digit number, $E$ and $G$ are digits.
    • $GG$ can be written as $G \times 111 = G \times 3 \times 37$.
    • So, $EF \times E = 3 \times 37 \times G$.
    • Since $37$ is prime, one of the factors on the left must be $37$ or a multiple of $37$.
    • Since $E$ is a single digit, the 2-digit number $EF$ must be $37$.
    • Thus, $E = 3$ and $F = 7$.
    • Check: $37 \times 3 = 111$. This matches $GGG$ with $G = 1$.
    • Answer: $E = 3, F = 7, G = 1$.

Practical Activities & Experiments

  • Digital Root Modular Wheel: Draw a circle with numbers $1$ to $9$ representing hours. Plot consecutive numbers on this wheel. Any natural number mapped to its digital root will fall on the same clock position as its remainder when divided by $9$ (with $9$ matching the $0$ position).
  • Parity Sign Swap Tally: Use a set of red (even) and blue (odd) cards. Write down an expression like $15 - 8 + 3 - 6$. Swap subtraction and addition signs arbitrarily, evaluating the expression each time to physically confirm that the parity remains constant.
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