GCF for 8, 21 and 75
What's the Greatest Common Factor (GCF) of 8, 21 and 75?
Answer
(One)
1 is the largest positive integer that divides 8, 21 and 75 without a remainder — every other common divisor of the three divides it too.
Finding the GCF of 8, 21 and 75 by Prime Factorization
Split all three numbers into prime factors, then keep every prime that appears in all three of them, each taken as many times as it appears in the poorest factorization. Multiplying what is left gives the greatest common factor:
All Prime Factors of 8: 2 × 2 × 2 (exponent form: 23)
All Prime Factors of 21: 3 × 7 (exponent form: 3 × 7)
All Prime Factors of 75: 3 × 5 × 5 (exponent form: 3 × 52)
| Prime factor | 8 | 21 | 75 | Lowest power |
|---|---|---|---|---|
| 2 | 23 | — | — | — |
| 3 | — | 3 | 3 | — |
| 5 | — | — | 52 | — |
| 7 | — | 7 | — | — |
Not one prime factor is shared by all three numbers, which makes 8, 21 and 75 relatively prime as a set. The only positive integer dividing all of them is 1.
Checking the GCF of 8, 21 and 75 Two Numbers at a Time
The greatest common factor is associative, so three numbers can be handled as two pairs — and this is what makes the Euclidean algorithm usable here: run it on the first pair, then run it again on that result and the third number.
GCF(a, b, c) = GCF(GCF(a, b), c)
GCF(8, 21) = 1
GCF(1, 75) = 1
GCF(8, 21, 75) = 1
The answer checks out by division — all three quotients come out whole:
8 ÷ 1 = 8
21 ÷ 1 = 21
75 ÷ 1 = 75
8, 21 and 75 have no common factor left — that is what makes 1 the greatest one.
Related Calculations
See Also
- Least Common Multiple of 3 Numbers - Find the Least Common Multiple (LCM) of three numbers
- Greatest Common Factor - Find the Greatest Common Factor (GCF) of two numbers
- Least Common Multiple - Find the Least Common Multiple (LCM) of two numbers
<a href="https://calculat.io/en/number/greatest-common-factor-of-3-numbers/8--21--75">What is the GCF of 8, 21 and 75? - Calculatio</a>
GCF Table
| Number 1 | Number 2 | Number 3 | GCF |
|---|---|---|---|
| 8 | 21 | 60 | 1 |
| 8 | 21 | 61 | 1 |
| 8 | 21 | 62 | 1 |
| 8 | 21 | 63 | 1 |
| 8 | 21 | 64 | 1 |
| 8 | 21 | 65 | 1 |
| 8 | 21 | 66 | 1 |
| 8 | 21 | 67 | 1 |
| 8 | 21 | 68 | 1 |
| 8 | 21 | 69 | 1 |
| 8 | 21 | 70 | 1 |
| 8 | 21 | 71 | 1 |
| 8 | 21 | 72 | 1 |
| 8 | 21 | 73 | 1 |
| 8 | 21 | 74 | 1 |
| 8 | 21 | 75 | 1 |
| 8 | 21 | 76 | 1 |
| 8 | 21 | 77 | 1 |
| 8 | 21 | 78 | 1 |
| 8 | 21 | 79 | 1 |
| 8 | 21 | 80 | 1 |
| 8 | 21 | 81 | 1 |
| 8 | 21 | 82 | 1 |
| 8 | 21 | 83 | 1 |
| 8 | 21 | 84 | 1 |
| 8 | 21 | 85 | 1 |
| 8 | 21 | 86 | 1 |
| 8 | 21 | 87 | 1 |
| 8 | 21 | 88 | 1 |
| 8 | 21 | 89 | 1 |
About "Greatest Common Factor of 3 Numbers" Calculator
The Greatest Common Factor of three numbers — also called the greatest common divisor (GCD) or highest common factor (HCF) — is the largest positive integer that divides all three of them with zero remainder