Lesson 11 of 1524
Number theory
Primes, factors, and the two number systems you will keep meeting.
Practice this chapterA prime has exactly two positive divisors, 1 and itself. The number 1 is neither prime nor composite. The number 2 is the only even prime.
The greatest common divisor is the largest whole number that divides both. The least common multiple is the smallest positive number both divide into. For two numbers, their product equals the GCD times the LCM.
GCD and LCM
a × b = GCD(a, b) × LCM(a, b)
If you know three of these, you can find the fourth.
Worked example
Find the GCD and LCM of 12 and 18.
- 1Factor 12: 12 = 2 × 6 = 2 × 2 × 3, which is 2² × 3.
- 2Factor 18: 18 = 2 × 9 = 2 × 3 × 3, which is 2 × 3².
- 3The GCD keeps the lower power of each shared prime. Both have a 2 and a 3, so GCD = 2 × 3 = 6.
- 4The LCM keeps the higher power of every prime that appears. That is 2² and 3², so LCM = 4 × 9 = 36.
- 5Check the product rule: 12 × 18 = 216, and GCD × LCM = 6 × 36 = 216.
Result: GCD 6, LCM 36
Why. 6 is the greatest whole number that divides both 12 and 18. 36 is the smallest positive number that both 12 and 18 divide into. Their product matches 12 × 18, which confirms the pair.
A rational number is a ratio of integers. √2 is real, but it is not rational.
Practice margin
This chapter
A fresh set from this chapter only. Choose 10 or 20. Multiple choice and fill-in, with no repeat inside the set.