A divisor common to two or more integers is called a “common divisor,” and a multiple common to them is called a “common multiple.”
Among these, the largest common divisor is called the greatest common divisor, and the smallest common multiple is called the least common multiple.
What You Will Learn
When finding the greatest common divisor and least common multiple, a method called the successive division method is very useful. It is also called the “sudare-zan” method.
The method is simple: divide two numbers successively by the same number until they can no longer be divided by the same number.
Example
Find the greatest common divisor and least common multiple of 12 and 18.
Solution:
① Using the successive division method, successively divide both numbers by numbers that divide them evenly.
2）12 18
3） 6 9
2 3 ← They can no longer be divided by the same number
② For the greatest common divisor, multiply all the “divisors on the left.”
2×3 = 6
③ For the least common multiple, multiply all the “divisors on the left” and the “numbers remaining at the bottom.”
2×3×2×3 = 36
Find the greatest common divisor and least common multiple of 12 and 18.
Solution:
① Using the successive division method, successively divide both numbers by numbers that divide them evenly.
2）12 18
3） 6 9
2 3 ← They can no longer be divided by the same number
② For the greatest common divisor, multiply all the “divisors on the left.”
2×3 = 6
③ For the least common multiple, multiply all the “divisors on the left” and the “numbers remaining at the bottom.”
2×3×2×3 = 36
Key Points
Remember
- Greatest common divisor = multiply only the “divisors” in the successive division method
- Least common multiple = multiply all the “divisors” and the “numbers remaining at the end” in the successive division method
- For two integers, “greatest common divisor × least common multiple = the product of the two numbers.”
Practice Problems
- Find the greatest common divisor and least common multiple of 24 and 36.
- Among numbers greater than 8, find a pair of integers whose greatest common divisor is 8 and whose least common multiple is 96.
- There are three ropes measuring 48 cm, 60 cm, and 84 cm. If you want to cut all of them into pieces of the same length, what should the length of each piece be to make the pieces as long as possible?
Answers and Explanations
- Answer: Greatest common divisor = 12, least common multiple = 72
Explanation: Using the successive division method: 2）24, 36 → 2）12, 18 → 3）6, 9 → 2, 3. Greatest common divisor = 2×2×3 = 12. Least common multiple = 2×2×3×2×3 = 72. - Answer: 24 and 32
Explanation: Since GCD×LCM = the product of the two numbers, 8×96 = 768. Let the two numbers be 8a and 8b, where a and b are relatively prime. Then 8ab = 96, so ab = 12. The relatively prime integer pair with a<b and a>1 is (a,b)=(3,4), giving 24 and 32. Check: GCD(24,32)=8 and LCM(24,32)=96 ✓. The pair (1,12) gives 8 and 96, but 8 is excluded because the numbers must be greater than 8. The pair (2,6) gives 16 and 48, but GCD(16,48)=16 ✗. The correct answer is 24 and 32 (check: GCD=8, LCM=96 ✓). - Answer: 12 cm
Explanation: To cut the three ropes into pieces of the same length and make the pieces as long as possible, find the greatest common divisor of the three lengths. Find GCD(48, 60, 84) using the successive division method: 2）48, 60, 84 → 2）24, 30, 42 → 3）12, 15, 21 → 4, 5, 7 (relatively prime). GCD = 2×2×3 = 12 cm.