Mathematics: Class 6 — Chapter 5: Prime Time

The Idli-Vada Game is a circle game of numbers where players say:

• 'idli' for multiples of 3 ($3, 6, 9, 12, \dots$)
• 'vada' for multiples of 5 ($5, 10, 15, 20, \dots$)
• 'idli-vada' for numbers that are multiples of both 3 and 5 ($15, 30, 45, \dots$)

Numbers that are multiples of both 3 and 5 are called common multiples. The first common multiple of 3 and 5 is $15$.

In the Jump Jackpot game, jumping on a grid helps identify factors. For example, jump sizes of 1, 2, 3, 4, 6, 8, 12, and 24 will land exactly on 24. These numbers are called factors or divisors of 24.

The jump sizes that can reach two different numbers are their common factors. For example, the common factors of 14 and 36 are 1 and 2.

A perfect number is a number for which the sum of all its factors is equal to twice the number. For example, 28 is a perfect number because its factors are 1, 2, 4, 7, 14, and 28, and their sum is $56 = 2 \times 28$. The only perfect number between 1 and 10 is 6.

The Sieve of Eratosthenes is a systematic method developed by the Greek mathematician Eratosthenes to find all prime numbers up to 100 by crossing out multiples of prime numbers starting from 2.

Twin primes are pairs of prime numbers that have a difference of 2. Examples include $(3, 5)$, $(5, 7)$, $(11, 13)$, $(17, 19)$, $(29, 31)$, $(41, 43)$, $(59, 61)$, and $(71, 73)$.

For any number, there is only one unique prime factorisation, regardless of the order of the factors. For example, the prime factorisation of 36 is always $2 \times 2 \times 3 \times 3$, which can be written as $2^2 \times 3^2$.

• Checking Co-primality: Two numbers are co-prime if they share no common prime factors in their prime factorisations. For example, $40 = 2 \times 2 \times 2 \times 5$ and $231 = 3 \times 7 \times 11$ share no common prime factors, so they are co-prime.
• Checking Divisibility: A number is divisible by another if the prime factorisation of the second number is completely included in the prime factorisation of the first number. For example, 168 ($2 \times 2 \times 2 \times 3 \times 7$) is divisible by 12 ($2 \times 2 \times 3$).

Numbers can have unique properties (e.g., 9 is the only single-digit square in a set, 43 is the only prime). Prime puzzles involve filling grids with prime numbers such that the product of each row and column matches the given target numbers.

• A number is a factor of another if it divides it exactly.
• Prime numbers have exactly two factors; composite numbers have more than two.
• Co-prime numbers share no common factors other than 1.
• Prime factorisation represents a number uniquely as a product of primes.