gcf (8,492; 1,641) = ? Calculate the greatest (highest) common factor (divisor) of numbers, gcf (hcf, gcd), by two methods: 1) The prime factorization and 2) The Euclidean Algorithm
gcf, hcf, gcd (8,492; 1,641) = ?
Method 1. The prime factorization:
The prime factorization of a number: finding the prime numbers that multiply together to make that number.
8,492 = 22 × 11 × 193
8,492 is not a prime number but a composite one.
1,641 = 3 × 547
1,641 is not a prime number but a composite one.
* The natural numbers that are only divisible by 1 and themselves are called prime numbers. A prime number has exactly two factors: 1 and itself.
* A composite number is a natural number that has at least one other factor than 1 and itself.
Calculate the greatest (highest) common factor (divisor):
Multiply all the common prime factors, taken by their smallest exponents (the smallest powers).
But the two numbers have no common prime factors.
The greatest (highest) common factor (divisor),
gcf, hcf, gcd (8,492; 1,641) = 1
Coprime numbers (prime to each other, relatively prime).
Scroll down for the 2nd method...
Method 2. The Euclidean Algorithm:
This algorithm involves the process of dividing numbers and calculating the remainders.
'a' and 'b' are the two natural numbers, 'a' >= 'b'.
Divide 'a' by 'b' and get the remainder of the operation, 'r'.
If 'r' = 0, STOP. 'b' = the gcf (hcf, gcd) of 'a' and 'b'.
Else: Replace ('a' by 'b') and ('b' by 'r'). Return to the step above.
Step 1. Divide the larger number by the smaller one:
8,492 ÷ 1,641 = 5 + 287
Step 2. Divide the smaller number by the above operation's remainder:
1,641 ÷ 287 = 5 + 206
Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
287 ÷ 206 = 1 + 81
Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
206 ÷ 81 = 2 + 44
Step 5. Divide the remainder of the step 3 by the remainder of the step 4:
81 ÷ 44 = 1 + 37
Step 6. Divide the remainder of the step 4 by the remainder of the step 5:
44 ÷ 37 = 1 + 7
Step 7. Divide the remainder of the step 5 by the remainder of the step 6:
37 ÷ 7 = 5 + 2
Step 8. Divide the remainder of the step 6 by the remainder of the step 7:
7 ÷ 2 = 3 + 1
Step 9. Divide the remainder of the step 7 by the remainder of the step 8:
2 ÷ 1 = 2 + 0
At this step, the remainder is zero, so we stop:
1 is the number we were looking for - the last non-zero remainder.
This is the greatest (highest) common factor (divisor).
The greatest (highest) common factor (divisor):
gcf, hcf, gcd (8,492; 1,641) = 1
Coprime numbers (prime to each other, relatively prime).
The two numbers have no prime factors in common
Why do we need to calculate the greatest common factor?
Once you've calculated the greatest common factor of the numerator and the denominator of a fraction, it becomes much easier to fully reduce (simplify) the fraction to the lowest terms (the smallest possible numerator and denominator).
Other similar operations with the greatest (highest) common factor (divisor):
The greatest (highest) common factor (divisor), gcf (hcf, gcd): the latest 5 calculated values
| The gcf, hcf, gcd (8,492 and 1,641) = ? | May 29 01:50 UTC (GMT) |
| The gcf, hcf, gcd (6,240 and 34) = ? | May 29 01:50 UTC (GMT) |
| The gcf, hcf, gcd (38 and 29,726) = ? | May 29 01:50 UTC (GMT) |
| The gcf, hcf, gcd (11 and 10) = ? | May 29 01:50 UTC (GMT) |
| The gcf, hcf, gcd (20,790 and 41,580,000) = ? | May 29 01:50 UTC (GMT) |
| The greatest (highest) common factor (divisor), gcf (hcf, gcd): the list of all the calculations |
Calculator of the greatest (highest) common factor (divisor), gcf, hcf, gcd
Calculate the greatest (highest) common factor (divisor) of numbers, gcd, hcf, gcd:
Method 1: Run the prime factorization of the numbers - then multiply all the common prime factors, taken by their smallest exponents. If there are no common prime factors, then gcf equals 1.
Method 2: The Euclidean Algorithm.
Method 3: The divisibility of the numbers.
The greatest (highest) common factor (divisor), gcf, hcf, gcd. What it is and how to calculate it.
- Note 1: The greatest common factor (gcf) is also called the highest common factor (hcf), or the greatest common divisor (gcd).
- Note 2: The Prime Factorization of a number: finding the prime numbers that multiply together to make that number.
- Suppose the number "t" evenly divides the number "a" ( = when evenly dividing the number "a" by "t", the remainder is zero).
- When we look at the prime factorization of "a" and "t", we find that:
- 1) all the prime factors of "t" are also prime factors of "a"
- and
- 2) the exponents of the prime factors of "t" are equal to or smaller than the exponents of the prime factors of "a" (see the * Note below)
- For example, the number 12 is a divisor (a factor) of the number 60:
- 12 = 2 × 2 × 3 = 22 × 3
- 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5
- * Note: 23 = 2 × 2 × 2 = 8. We say that 2 was raised to the power of 3. In this example, 3 is the exponent and 2 is the base. The exponent indicates how many times the base is multiplied by itself. 23 is the power and 8 is the value of the power.
- If the number "t" is a common divisor of the numbers "a" and "b", then:
- 1) "t" only has the prime factors that also intervene in the prime factorization of "a" and "b".
- and
- 2) each prime factor of "t" has the smallest exponents with respect to the prime factors of the numbers "a" and "b".
- For example, the number 12 is the common divisor of the numbers 48 and 360. Below is their prime factorization:
- 12 = 22 × 3
- 48 = 24 × 3
- 360 = 23 × 32 × 5
- You can see that the number 12 has only the prime factors that also occur in the prime factorization of the numbers 48 and 360.
- You can see above that the numbers 48 and 360 have several common factors: 2, 3, 4, 6, 8, 12, 24. Out of these, 24 is the greatest common factor (GCF) of 48 and 360.
- 24 = 2 × 2 × 2 × 3 = 23 × 3
- 48 = 24 × 3
- 360 = 23 × 32 × 5
- 24, the greatest common factor of the numbers 48 and 360, is calculated as the product of all the common prime factors of the two numbers, taken by the smallest exponents (powers).
- If two numbers "a" and "b" have no other common factor than 1, gcf (a, b) = 1, then the numbers "a" and "b" are called coprime numbers (relatively prime, prime to each other).
- If "a" and "b" are not relatively prime numbers, then every common divisor of "a" and "b" is a divisor of the greatest common divisor of "a" and "b".
- Let's have an example on how to calculate the greatest common factor, gcf, of the following numbers:
- 1,260 = 22 × 32
- 3,024 = 24 × 32 × 7
- 5,544 = 23 × 32 × 7 × 11
- gcf (1,260, 3,024, 5,544) = 22 × 32 = 252
- And another example:
- 900 = 22 × 32 × 52
- 270 = 2 × 33 × 5
- 210 = 2 × 3 × 5 × 7
- gcf (900, 270, 210) = 2 × 3 × 5 = 30
- And one more example:
- 90 = 2 × 32 × 5
- 27 = 33
- 22 = 2 × 11
- gcf (90, 27, 22) = 1 - The three numbers have no prime factors in common, they are relatively prime.
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