5
Prime and composite numbers
Textbook, pp. 16–18
GoalTell prime from composite numbers; find primes with the sieve of Eratosthenes.
New words
prime number · tub soncomposite number · murakkab sonsieve of Eratosthenes · Eratosfen g‘alviridivisor · bo‘luvchi
Explanation
A natural number with exactly two divisors, 1 and itself, is a prime number. A natural number with more than two divisors is composite. The number 1 is neither prime nor composite because it has only one divisor. The number 2 is the only even prime; every other even number is divisible by 2 and so is composite. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. The ancient Greek scholar Eratosthenes suggested finding primes by crossing out composite numbers from a table one after another.
Worked examples
17 is prime: it divides only by 1 and 17. 21 is composite: 21 = 3 · 7, with divisors 1, 3, 7, 21.
Sieve: keep 2 and cross out its multiples; then keep 3 and cross out its multiples, and so on. The numbers left are prime.
Class activity
“Living sieve”: cards with numbers 2 to 50 are handed out. The teacher says “multiples of 2, sit down!”, then of 3, 5 and 7. Those still standing at the end show the primes.
Practice
1
Which of 21, 23, 27, 33 is prime?
23
2
List all primes between 20 and 40.
23, 29, 31, 37
3
How many primes are there from 1 to 20?
8: 2, 3, 5, 7, 11, 13, 17, 19.
4
Show that 91 is composite.
91 = 7 · 13, so it has divisors other than 1 and 91.