Common Multiples and Factors

Common Multiples

In the Idli-Vada game, we saw that some numbers belong to two groups at once.

• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30…
• Multiples of 5: 5, 10, 15, 20, 25, 30…

The numbers 15, 30, 45… are multiples of both 3 and 5. These are called Common Multiples.

Factors and Divisors

A factor (or divisor) is a number that divides another number exactly (leaving no remainder).

Jump Jackpot Example

Imagine a game where you must jump in steps of a specific size to land exactly on a treasure at number 24.

• Jump size 4? Yes: $4 \to 8 \to 12 \to 16 \to 20 \to 24$.
• Jump size 5? No: $5 \to 10 \to 15 \to 20 \to 25$ (Missed it!).

The successful jump sizes are the factors of 24: $\text{Factors of 24: } 1, 2, 3, 4, 6, 8, 12, 24$

Common Factors

If treasures are kept on 14 and 36, which jump size lands on both?

• Factors of 14: 1, 2, 7, 14
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Common Factors: 1, 2.

Perfect Numbers

A number is called a Perfect Number if the sum of all its factors is equal to twice the number.

Example: 28

• Factors: 1, 2, 4, 7, 14, 28
• Sum: $1 + 2 + 4 + 7 + 14 + 28 = 56$
• Check: $28 \times 2 = 56$.
• Yes, 28 is a perfect number!

Example: 6

• Factors: 1, 2, 3, 6
• Sum: $1 + 2 + 3 + 6 = 12$
• Check: $6 \times 2 = 12$.
• Yes, 6 is a perfect number!