Calculate a greatest common factor
Keep the default values for 210 and 90, or compare any two whole numbers.
Example data
| First number | Second number | Greatest common factor | Least common multiple |
|---|---|---|---|
| 210 | 90 | 30 | 630 |
| 84 | 126 | 42 | 252 |
| 72 | 120 | 24 | 360 |
| 45 | 64 | 1 | 2,880 |
Formula used
Euclidean algorithm: gcd(a, b) = gcd(b, a mod b).
Factor rule: retain only shared prime factors, using each smaller exponent.
The Euclidean algorithm repeatedly divides the larger value by the smaller value. Continue with the divisor and remainder. The last nonzero remainder is the greatest common factor. For 210 and 90, the divisions end with 30. Therefore, 30 is their GCF.
How to use this calculator
- Enter the first whole number in the first input.
- Enter the second whole number in the next input.
- Select full steps or a short result summary.
- Press Find the GCF to place results above the form.
- Use CSV or PDF buttons after calculating to save the result.
Understanding the greatest common factor
The greatest common factor, or GCF, is the largest positive integer that divides two whole numbers exactly. It helps simplify fractions, group objects evenly, and compare number patterns. For 210 and 90, the GCF is 30. Both numbers divide by 30 with no remainder.
Why 30 is the answer
Start with prime factorization. The number 210 equals 2 × 3 × 5 × 7. The number 90 equals 2 × 3² × 5. Their shared prime factors are 2, 3, and 5. Multiplying them gives 30. A larger shared product cannot be formed because 90 has only one factor of 3.
Using Euclidean division
The Euclidean method is often faster for larger numbers. Divide 210 by 90. The remainder is 30. Then divide 90 by 30. The remainder is zero. The last nonzero divisor is 30. This method avoids listing every factor, so it works well when numbers become large.
Useful related results
Every common factor of two numbers must divide their GCF. The common factors of 210 and 90 are 1, 2, 3, 5, 6, 10, 15, and 30. The least common multiple is 630. The GCF and LCM connect through this rule: GCF × LCM equals the absolute product of both numbers.
Where the GCF helps
Use the GCF to reduce fractions. For example, 90/210 becomes 3/7 after dividing both parts by 30. It also helps create equal groups. You could organize 210 and 90 items into groups of 30 without leftovers. In algebra, factoring out a GCF makes expressions shorter and clearer.
Good checking habits
Check that the reported GCF divides both input numbers. Next, test whether a larger candidate works. If it fails for either number, it cannot be the answer. Negative signs do not change the positive GCF. Zero has special handling: the GCF of zero and a nonzero number is the absolute nonzero number.
Make number work easier
This calculator shows the answer and the reasoning. Use full mode when you want divisors, prime factors, and division steps. Use summary mode when you only need the key values. Save a CSV for spreadsheet work, or create a PDF for a clean record. Practice with different pairs to strengthen factor recognition. Practice builds confidence with fractions, ratios, and algebraic expressions. It also supports everyday grouping tasks with larger values.
Frequently asked questions
1. What is the GCF of 210 and 90?
The GCF is 30. It is the largest positive whole number that divides both 210 and 90 without leaving a remainder.
2. How can I check the answer?
Divide both numbers by the proposed GCF. Here, 210 ÷ 30 equals 7 and 90 ÷ 30 equals 3. Both quotients are whole numbers.
3. Is GCF the same as HCF?
Yes. GCF means greatest common factor. HCF means highest common factor. Both names describe the same largest shared divisor.
4. Why does the calculator show the LCM?
The LCM is a useful related value. It is the smallest positive multiple shared by both numbers. For 210 and 90, it equals 630.
5. Can negative numbers be used?
Yes. The calculator uses absolute values when finding the GCF. For example, the GCF of -210 and 90 is still 30.
6. What happens when one input is zero?
The GCF of zero and a nonzero whole number is the absolute value of the nonzero number. Both inputs cannot be zero here.
7. Why are prime factors useful?
Prime factors reveal the building blocks of each number. Shared primes, taken with their smallest powers, multiply to form the GCF.
8. What is the Euclidean algorithm?
It repeatedly replaces a number pair with the divisor and remainder. The last nonzero remainder identifies the GCF efficiently.
9. Can the GCF be larger than either input?
No. A common factor must divide each input. Therefore, it cannot be larger than the smaller nonzero input value.
10. Does this calculator work for coprime numbers?
Yes. Coprime numbers share no positive factor except 1. The calculator returns 1 as their greatest common factor.
11. How do CSV and PDF downloads help?
CSV creates a simple data file for spreadsheets. PDF creates a readable calculation summary that you can save, print, or share.