Unit 2 · Topic 03 · Number Play
Anaya said 8 and 9 had nothing in common. Kabir said that was impossible — every pair of numbers shares at least one factor.
Both of them were right, in a way neither had expected.
It took Ms. Rao one shared factor to show they were arguing about two completely different meanings of "nothing in common."
"8 and 9 have nothing in common," said Anaya, comparing her two locker numbers. Kabir disagreed instantly. "That's not possible — every number is divisible by 1, so 1 is always a common factor. They always have SOMETHING in common." Anaya insisted she wasn't wrong either. They brought it to Ms. Rao, both certain the other had made an error.
Ms. Rao listed the factors of 8: 1, 2, 4, 8. Then the factors of 9: 1, 3, 9. "Kabir's right that 1 appears in both lists — every pair of whole numbers shares 1 as a factor. But Anaya, what did you actually mean by 'nothing in common'?" Anaya said she meant no factor OTHER than 1. Comparing the two lists again, that was true: apart from the shared 1, nothing else matched.
"You were both using the phrase correctly, just meaning different things by it," said Ms. Rao. "When two numbers share no factor except 1, mathematicians call them co-prime. It doesn't mean either number is prime by itself — 8 and 9 are both composite — it only means the PAIR shares nothing beyond 1." Kabir tested a few more pairs: 6 and 9 share 1 and 3, so they are NOT co-prime. But 6 and 25 share only 1, so they ARE co-prime, even though neither 6 nor 25 is prime.
Ms. Rao then introduced a second idea: twin primes, two prime numbers exactly 2 apart, like 3 and 5, or 11 and 13. "Twin primes are automatically co-prime too," she pointed out, "since two different primes can never share any factor except 1." Anaya checked 17 and 19 — both prime, two apart, and their only shared factor was 1, confirming both ideas fit together neatly.
Two numbers are co-prime (or relatively prime) if their only common factor is 1 — even if neither number is itself prime, like 8 and 9, or 6 and 25.
Co-prime is a property of a PAIR, not a single number. A number is prime or composite on its own, but co-prime only makes sense when comparing two numbers together.
Twin primes are two prime numbers that differ by exactly 2, like (3,5), (5,7), (11,13), (17,19). Since two different primes only ever share the factor 1, every pair of twin primes is automatically co-prime too.
List the factors of any two numbers under 20 you pick yourself. Compare the lists and decide if the pair is co-prime.
Search a hundred-number chart (1-100) for at least three pairs of twin primes besides the ones shown here.
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