Co-prime Numbers

What are Co-prime Numbers?

Two numbers are said to be co-prime if they have no common factor other than 1.

Example: 4 and 9

• Factors of 4: 1, 2, 4
• Factors of 9: 1, 3, 9
• Common Factors: Only 1.
• Conclusion: 4 and 9 are co-prime.

Example: 15 and 39

• Factors of 15: 1, 3, 5, 15
• Factors of 39: 1, 3, 13, 39
• Common Factors: 1, 3.
• Conclusion: Not co-prime (because they share 3).

Co-prime Art (Thread Art)

If you arrange pegs in a circle and connect them with a thread using a specific gap:

• If the total number of pegs and the gap size are co-prime, the thread will touch every peg.
• If they share a common factor, the thread will form a smaller loop and miss some pegs.

Visualizing Thread Art (12 pegs, gap 4): 12 and 4 share a factor (4). The thread hits 12, 4, 8, then back to 12. It misses most pegs.

Visualizing Thread Art (13 pegs, gap 3): 13 is prime, so 13 and 3 are co-prime. The thread will eventually touch every single peg.

Properties of Co-primes in Multiplication

When playing the Idli-Vada game with co-prime numbers (like 3 and 5):

• The first Common Multiple is exactly the Product of the two numbers ($3 \times 5 = 15$).

If numbers are NOT co-prime (like 4 and 6, common factor 2):

• The first Common Multiple (12) is less than their product ($4 \times 6 = 24$).