7
Greatest common divisor. Coprime numbers
Textbook, pp. 21–25
GoalFind the greatest common divisor (GCD); understand coprime numbers.
New words
common divisor · umumiy bo‘luvchigreatest common divisor (GCD) · EKUBcoprime numbers · o‘zaro tub sonlarprime factors · tub ko‘paytuvchilar
Explanation
A number that divides both of two numbers exactly is a common divisor of them. The greatest of the common divisors is the greatest common divisor (GCD). To find it, factor the numbers into primes, take the prime factors that appear in both with their smallest powers and multiply. For example 24 = 2³ · 3 and 36 = 2² · 3², so the common part is 2² · 3 = 12. If the GCD of two numbers is 1, they are coprime; the numbers themselves need not be prime, for example 14 and 25.
Worked examples
GCD(48, 60): 48 = 2⁴ · 3, 60 = 2² · 3 · 5. The common part is 2² · 3 = 12.
18 and 35: 18 = 2 · 3², 35 = 5 · 7. No common prime factor, GCD = 1, so they are coprime.
Class activity
“Biggest boxes”: groups get 36 red and 48 blue pencils (cards or counting sticks). They must split them into identical sets: what is the greatest number of sets? Groups check their answer with the GCD.
Practice
1
Find GCD(48, 60).
12
2
Are 14 and 25 coprime?
Yes, GCD(14, 25) = 1, even though both are composite.
3
36 red and 48 blue pencils are split into identical sets using all pencils. What is the greatest number of sets?
12
4
Why is GCD(7, 21) = 7?
21 is divisible by 7, so the smaller number divides the bigger one and is itself the GCD.