Co-prime numbers

Common factors in the treasure game

In the Jump Jackpot game, Jumpy starts at 0 and jumps by a fixed size; he wins if he lands on the treasures. To land on two treasures, his jump size must be a factor of both numbers, that is, a common factor. For treasures on 28 and 70, the jump sizes that work are 1, 2, 7 and 14.

Safe pairs and co-prime numbers

Now a new rule: a jump size of 1 is not allowed. Grumpy wants to place two treasures so that Jumpy cannot reach both. Placing them on 12 and 26 fails, because a jump size of 2 reaches both. But 4 and 9 are safe: no jump size other than 1 reaches both.

Safe pairs have no common factor other than 1. Such numbers are called co-prime. Checking the book’s pairs:

Pair	Co-prime?
15 and 39	No: common factor 3
4 and 15	Yes
18 and 29	Yes
20 and 55	No: common factor 5
18 and 35	Yes
30 and 415	No: common factor 5
17 and 69	Yes (69 = 3 × 23)
81 and 18	No: common factor 9

Co-prime does not mean prime

Co-prime describes a pair of numbers. The numbers themselves can be composite: 4 = 2 × 2 and 9 = 3 × 3 share no factor except 1, so they are co-prime. Two different primes are always co-prime, because each has only 1 and itself as factors.

Checking with prime factorisation

Finding one factor pair is not enough. Anshu wrote 56 = 14 × 4 and 63 = 21 × 3, saw no common factor and wrongly said they were co-prime; Guna wrote 56 = 7 × 8 and 63 = 9 × 7 and found 7. The reliable way is the full prime factorisation:

• 56 = 2 × 2 × 2 × 7 and 63 = 3 × 3 × 7 share the prime 7, so they are not co-prime.
• 40 = 2 × 2 × 2 × 5 and 231 = 3 × 7 × 11 share no prime, so they are co-prime.
• 242 = 2 × 11 × 11 and 195 = 3 × 5 × 13 share no prime, so they are co-prime.

If two numbers had a composite common factor, its prime factors would appear in both factorisations, so no common prime means no common factor except 1.

The first common multiple

In the idli-vada game, Anshu noticed that the first common multiple of two numbers is sometimes equal to their product and sometimes smaller. For co-prime pairs it equals the product: 4 and 9 first meet at 36 = 4 × 9, and 3 and 5 at 15. When the numbers share a factor, it is smaller: 4 and 6 first meet at 12, not 24, and 6 and 8 at 24, not 48.

Co-prime thread art

Tie a thread round a circle of pegs, skipping a fixed number each time. The thread reaches every peg only when the number of pegs and the thread-gap are co-prime.

For more on breaking numbers into primes, read prime factorisation. For the textbook’s questions, see the Chapter 5 NCERT solutions; for the whole chapter, see the Chapter 5 notes.