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Máy tính ước chung lớn nhất và bội chung nhỏ nhất

Greatest common divisor (GCD, GCF or HCF) and least common multiple (LCM) of two or more whole numbers, with Euclid's algorithm step by step.

Cập nhật Ví dụ đã kiểm tra: 6

Two or more integers, separated by commas or spaces.
Thử
Greatest common divisor
Greatest common divisor: 6
Số nguyên; Gần nhất; nếu cách đều thì ra xa số không
Least common multiple
180

The largest number dividing 12, 18 and 30 exactly is 6; the smallest positive number they all divide into is 180.

Euclid's algorithm as squares: 12 × 18 rectangle

1261812
Euclid's algorithm, a = q·b + r Số hàng: 3
PairabQuotient qRemainder r
gcd(12, 18)181216
gcd(12, 18)12620
gcd(6, 30)30650
Cách tính S

  1. Euclid's algorithm for gcd(12, 18)

    18=1×12+612=2×6+0⇒gcd⁡=6\begin{aligned}18 &= 1 \times 12 + 6 \\ 12 &= 2 \times 6 + 0\end{aligned} \qquad \Rightarrow \gcd = 6

    Divide, keep the remainder, repeat with the divisor; the last non-zero remainder is the gcd.

  2. Least common multiple with 18

    lcm⁡(12,18)=12×18gcd⁡(12,18)=2166=36\operatorname{lcm}(12, 18) = \frac{12 \times 18}{\gcd(12, 18)} = \frac{216}{6} = 36

  3. Euclid's algorithm for gcd(6, 30)

    30=5×6+0⇒gcd⁡=6\begin{aligned}30 &= 5 \times 6 + 0\end{aligned} \qquad \Rightarrow \gcd = 6

    Divide, keep the remainder, repeat with the divisor; the last non-zero remainder is the gcd.

  4. Least common multiple with 30

    lcm⁡(36,30)=36×30gcd⁡(36,30)=1,0806=180\operatorname{lcm}(36, 30) = \frac{36 \times 30}{\gcd(36, 30)} = \frac{1{,}080}{6} = 180

Giới thiệu Máy tính ước chung lớn nhất và bội chung nhỏ nhất

The greatest common divisor (GCD, also called the greatest common factor or highest common factor) is the largest whole number that divides every input; the least common multiple (LCM) is the smallest positive number that every input divides. Euclid's algorithm finds the GCD by repeatedly replacing the larger number with its remainder after division by the smaller: gcd(1071, 462) = gcd(462, 147) = gcd(147, 21) = 21. The LCM follows from lcm(a, b) = |a × b| ÷ gcd(a, b), applied a pair at a time.

Reducing fractions uses the GCD; adding fractions and lining up repeating schedules use the LCM. For the default 12, 18 and 30 the GCD is 6 and the LCM is 180, so events every 12, 18 and 30 days next coincide after 180 days.

For two numbers the calculator also gives the Bézout identity from the extended Euclidean algorithm: 21 = 1071 × (−3) + 462 × 7. Negative inputs count by their absolute value, gcd(0, n) = n, and a list containing 0 has an LCM of 0.

Ví dụ có lời giải

12, 18 and 30

Whole numbers
12, 18, 30
Greatest common divisor
6
Least common multiple
180

Nguồn đối chiếu: Python 3.8 math.gcd and a·b // gcd folded over the list

Euclid's 1071 and 462

Whole numbers
1071, 462
Greatest common divisor
21
Least common multiple
23,562
Bézout identity
21 = 1071·(−3) + 462·7

Nguồn đối chiếu: Wikipedia — Euclidean algorithm worked example (gcd 21); lcm and Bézout coefficients by hand back-substitution, checked in Python

Extended Euclid on 240 and 46

Whole numbers
240 46
Greatest common divisor
2
Least common multiple
5,520
Bézout identity
2 = 240·(−9) + 46·47

Nguồn đối chiếu: Wikipedia — Extended Euclidean algorithm example table (s = −9, t = 47)

Coprime 17 and 31

Whole numbers
17, 31
Greatest common divisor
1
Least common multiple
527

Nguồn đối chiếu: Both prime, so gcd 1 and lcm 17 × 31 = 527

Câu hỏi

How do you find the GCD of two numbers?

Use Euclid's algorithm: divide the larger number by the smaller, keep the remainder, and repeat with the divisor and the remainder until the remainder is 0; the last non-zero remainder is the GCD. For 1071 and 462: 1071 = 2 × 462 + 147, 462 = 3 × 147 + 21, 147 = 7 × 21 + 0, so the GCD is 21. By Lamé's theorem it never needs more than five steps per digit of the smaller number.

How do you find the LCM of two numbers?

Divide their product by their GCD: lcm(a, b) = a × b ÷ gcd(a, b). For 12 and 18, gcd = 6, so lcm = 216 ÷ 6 = 36. For more numbers, go one pair at a time: lcm(36, 30) = 1,080 ÷ 6 = 180, so the LCM of 12, 18 and 30 is 180. Listing multiples (12, 24, 36 …) reaches the same answer, but slowly for large numbers.

What is the difference between GCD, GCF and HCF?

There is none: greatest common divisor (GCD), greatest common factor (GCF) and highest common factor (HCF) are three names for the same number. GCF and HCF are the usual school terms, GCF mostly in the US and HCF in the UK and India, while GCD is the name in number theory and programming, as in Python's math.gcd. Under every name, gcd(12, 18) = 6.

How are the GCD and LCM related?

For two positive whole numbers, gcd(a, b) × lcm(a, b) = a × b; with 12 and 18, 6 × 36 = 216 = 12 × 18. In prime factors, the GCD takes the lower power of each shared prime and the LCM the higher power of every prime: 12 = 2² × 3 and 18 = 2 × 3², so the GCD is 2 × 3 = 6 and the LCM is 2² × 3² = 36. The product rule does not extend to three or more numbers.

What does it mean when two numbers are coprime?

Their GCD is 1, so they share no prime factor, and their LCM is simply their product. 17 and 31 are coprime, with an LCM of 17 × 31 = 527, and so are 8 and 15 although neither is prime. A fraction is in lowest terms exactly when its numerator and denominator are coprime.

“Máy tính ước chung lớn nhất và bội chung nhỏ nhất” chính xác đến mức nào?

Độ chính xác phụ thuộc vào dữ liệu nhập và giả định của phương pháp. Phép tính thập phân dùng 50 chữ số có nghĩa, nhưng ước lượng, phương pháp số và dữ liệu nguồn có thể kém chính xác hơn; làm tròn khi hiển thị không loại bỏ các giới hạn đó. Ví dụ có lời giải đã đối chiếu với nguồn độc lập: 6. Ví dụ, “12, 18 and 30” được kiểm tra bằng Python 3.8 math.gcd and a·b // gcd folded over the list.

Phương pháp này lấy từ đâu?

Euclid, Elements, Book VII, Propositions 1–2; Knuth, The Art of Computer Programming Vol. 2, §4.5.2 (Euclid's algorithm); Wikipedia — Extended Euclidean algorithm (worked example 240, 46).

Về công cụ tính này

gcd⁡(a,b)=gcd⁡(b, a mod b)gcd⁡(a,0)=∣a∣lcm⁡(a,b)=∣a b∣gcd⁡(a,b)\begin{gathered} \gcd(a, b) = \gcd(b,\ a \bmod b) \\[6pt] \gcd(a, 0) = |a| \\[10pt] \operatorname{lcm}(a, b) = \frac{|a\,b|}{\gcd(a, b)} \end{gathered}

Nguồn

  1. Euclid, Elements, Book VII, Propositions 1–2
  2. Knuth, The Art of Computer Programming Vol. 2, §4.5.2 (Euclid's algorithm)
  3. Wikipedia — Extended Euclidean algorithm (worked example 240, 46)

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