# اعظم مشترک مقسوم علیہ اور اقل مشترک مضرب کا کیلکولیٹر

> 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.

انٹرایکٹو صورت: https://www.calcopenly.com/ur/math/gcd-lcm-calculator
موضوع: ریاضی کے کیلکولیٹر

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.

## اندراجات

- **Whole numbers**: Two or more integers, separated by commas or spaces.

## نتائج

- Greatest common divisor — بنیادی نتیجہ
- Least common multiple
- Bézout identity

## فارمولا

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

## حل شدہ مثالیں

### 12, 18 and 30

- Whole numbers: 12, 18, 30
- **Greatest common divisor: 6**
- **Least common multiple: 180**
- جانچ کا ماخذ: ⁨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**
- جانچ کا ماخذ: ⁨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**
- جانچ کا ماخذ: ⁨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**
- جانچ کا ماخذ: ⁨Both prime, so gcd 1 and lcm 17 × 31 = 527⁩

### Zero with 5 (edge)

- Whole numbers: 0, 5
- **Greatest common divisor: 5**
- **Least common multiple: 0**
- **Bézout identity: 5 = 0·0 + 5·1**
- جانچ کا ماخذ: ⁨gcd(0, n) = n and lcm(0, n) = 0 by definition (Python math.gcd(0, 5) = 5)⁩

### Negative input −12 and 18

- Whole numbers: -12, 18
- **Greatest common divisor: 6**
- **Least common multiple: 36**
- جانچ کا ماخذ: ⁨Python math.gcd(-12, 18) = 6; lcm = |−12·18| / 6⁩

## سوالات

### 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.

### “⁨اعظم مشترک مقسوم علیہ اور اقل مشترک مضرب کا کیلکولیٹر⁩” کتنا درست ہے؟

درستی آپ کی درج کردہ قدروں اور طریقے کے مفروضوں پر منحصر ہے۔ اعشاری حساب 50 بامعنی ہندسے استعمال کرتا ہے، مگر تخمینے، عددی طریقے اور ماخذ کا ڈیٹا کم درست ہو سکتے ہیں؛ دکھائی گئی قدروں کو راؤنڈ کرنے سے یہ حدود ختم نہیں ہوتیں۔ آزاد ذرائع کی حل شدہ مثالوں سے جانچ: 6۔ مثلاً، “⁨12, 18 and 30⁩” کو ⁨Python 3.8 math.gcd and a·b // gcd folded over the list⁩ سے جانچا جاتا ہے۔

### اس طریقے کا ماخذ کیا ہے؟

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).

## ماخذ

- 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)](https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm)