Idli-Vada Game PRIME TIME

Chapter at a Glance

This chapter covers multiples, common multiples, factors, common factors, and divisibility. Using the "Idli-Vada" circle game and a "Jump Jackpot" treasure hunt, it introduces the core properties of numbers. Students learn to classify numbers as prime, composite, or neither, and explore the Sieve of Eratosthenes. The chapter details co-prime numbers, prime factorization, and tests of divisibility (for 2, 4, 5, 8, and 10), concluding with mathematical puzzles and co-prime circle thread art.

Key Definitions & Terminology

• Multiple: The product of a given number and any whole number.
• Common Multiple: A number that is a multiple of two or more given numbers.
• Factor (or Divisor): A whole number that divides another number exactly, leaving a remainder of zero.
• Common Factor: A number that divides two or more given numbers exactly.
• Perfect Number: A positive integer that is equal to the sum of all its proper factors (excluding the number itself). Alternatively, the sum of all its factors is twice the number (e.g. 6 and 28).
• Prime Number: A whole number greater than 1 that has exactly two factors: 1 and the number itself.
• Composite Number: A whole number greater than 1 that has more than two factors.
• Twin Primes: Pairs of prime numbers that have a difference of exactly 2 (e.g. 3 and 5).
• Co-prime Numbers: A pair of numbers that have no common factors other than 1.
• Prime Factorisation: The representation of a composite number as a product of prime numbers.
• Sieve of Eratosthenes: An ancient Greek grid-filtering algorithm used to find all prime numbers up to a given limit.

Formulas, Rules & Properties

• The Fundamental Theorem of Arithmetic:
  Every composite number greater than 1 can be uniquely factored into a product of prime numbers, ignoring the order of the factors.
• LCM of Co-prime Numbers:
  If two numbers are co-prime, their least common multiple (LCM) is equal to their product. If they are not co-prime, their LCM is strictly less than their product.
• Sophie Germain Primes (Primes $p$ where $2p + 1$ is also prime):
  Examples include: $2 \rightarrow 5$, $3 \rightarrow 7$, $5 \rightarrow 11$, $11 \rightarrow 23$, $23 \rightarrow 47$, $29 \rightarrow 59$, $41 \rightarrow 83$.
• Divisibility Test Rules:
• Divisibility by 2: Units digit must be even ($0, 2, 4, 6, 8$).
• Divisibility by 5: Units digit must be $0$ or $5$.
• Divisibility by 10: Units digit must be $0$.
• Divisibility by 4: The number formed by the last two digits must be divisible by 4.
• Divisibility by 8: The number formed by the last three digits must be divisible by 8.
• Co-prime Product Divisibility:
  If a number is divisible by two co-prime numbers $a$ and $b$, it is divisible by their product $ab$.

Core Concepts & Topics

• Idli-Vada Game:
• Multiples of 3 $\rightarrow$ "idli"
• Multiples of 5 $\rightarrow$ "vada"
• Multiples of both (i.e. multiples of 15) $\rightarrow$ "idli-vada".
• Jump Jackpot (Factors and Divisors):
• Starting at 0, a constant jump size $k$ lands on treasure $N$ if and only if $k$ is a factor of $N$.
• Landing on two treasures $M$ and $N$ requires the jump size to be a common factor of both.
• Sieve of Eratosthenes Method:
• In a 1–100 grid: Cross out 1. Circle 2 and cross out all its multiples. Circle the next uncrossed number (3) and cross out its multiples. Repeat until all numbers are processed. Circling identifies primes; crossing out (except 1) identifies composites.
• Co-prime Thread Art:
• In a circle of $P$ pegs, tying a thread to every $G$-th peg (thread-gap $G$) touches every single peg if and only if $P$ and $G$ are co-prime ($\gcd(P, G) = 1$). If they share a common factor, the thread loops back early, missing some pegs.
• Prime Factorisation Divisibility Check:
• $A$ is divisible by $B$ if the prime factorization of $B$ is fully included in the prime factorization of $A$ (matching base primes and their powers).

Worked Examples

• Idli-Vada Game Counts:
• Question: For numbers 1 to 90, how many times do players say each term?
  Solution:
  • "idli" (multiples of 3): $90 / 3 = \mathbf{30}$ times.
  • "vada" (multiples of 5): $90 / 5 = \mathbf{18}$ times.
  • "idli-vada" (multiples of 15): $90 / 15 = \mathbf{6}$ times.
• Idli-Vada game without vada:
• Question: In a game, players say only "idli" or "idli-vada" (no one says "vada"). One number is 4. What could the other number be from ${2, 3, 5, 8, 10}$?
  Solution: If no one says just "vada", the smaller number must divide the larger number. Since the smaller number is 4, the larger number must be a multiple of 4. The only choice is 8.
• Smallest Multiples of 1–10:
• Question: Find the smallest positive integer that is a multiple of all numbers from 1 to 10 (a) excluding 7, (b) including 7.
  Solution:
  • Excluding 7: $\text{LCM}(1, 2, 3, 4, 5, 6, 8, 9, 10) = 2^3 \times 3^2 \times 5 = \mathbf{360}$.
  • Including 7: $\text{LCM}(1, 2, \dots, 10) = 360 \times 7 = \mathbf{2520}$.
• Sophie Germain Product:
• Question: Find three prime numbers less than 30 whose product is 1955.
  Solution: Factoring 1955: $1955 = 5 \times 391 = 5 \times 17 \times 23$. The primes are $5, 17, 23$.
• Three-Digit Prime from ${2, 4, 5}$:
• Question: How many 3-digit prime numbers can you make using digits 2, 4, and 5 once?
  Solution: All permutations ($245, 254, 425, 452, 524, 542$) are either divisible by 2 (ending in an even digit) or by 5 (ending in 5). Thus, the answer is none (0).
• Finding Factors without Multiplying:
• Question: Find the prime factorization of $56 \times 25$.
  Solution: $56 = 2 \times 2 \times 2 \times 7$ and $25 = 5 \times 5$. Putting them together: $2 \times 2 \times 2 \times 5 \times 5 \times 7$.
• Divisibility Shortcut:
• Question: Guna checks a number for divisibility by only two of ${2, 4, 5, 8, 10}$ and proves it is divisible by all of them. What are the two?
  Solution: Checking for 5 and 8 is sufficient. Divisibility by 8 covers 2, 4, and 8. Divisibility by 5 covers 5. Since 5 and 8 are co-prime, divisibility by both implies divisibility by $5 \times 8 = 40$ (which covers 10).
• Factors of 10,000 ending in non-zero digits:
• Question: Write two numbers whose product is 10,000, where neither number ends in 0.
  Solution: $10000 = 2^4 \times 5^4 = \mathbf{16 \times 625}$.
• Perfect Number Search:
• Question: Find a perfect number between 1 and 10.
  Solution: For the number $6$, the factors are $1, 2, 3, 6$.
  $$\text{Sum} = 1 + 2 + 3 + 6 = 12 = 2 \times 6 \quad (\text{Perfect number rule holds}).$$

Practical Activities & Experiments

• Classroom Idli-Vada Game: Playing in a circle to practice multiples of 3, 5, and 15.
• Sieve of Eratosthenes Grid: Constructing a 100-number table and coloring primes vs. composites.
• Thread Peg Patterns: Winding thread around a pegboard to observe co-prime Peg-Gap relationships.
• Prime factor tree building: Drawing decomposition trees for numbers like 36 and 72.
• Prime Crossword Grid: Solving grid puzzles using prime factor products.