gcf, hcf, gcd (240; 160) = ? Calculate the greatest (highest) common factor (divisor) GCF (HCF, GCD), using the online calculator

Calculate the greatest (highest) common factor (divisor), gcf, hcf, gcd (240; 160) = ? Approach 1. Integer numbers prime factorization. Approach 2. Euclid's algorithm.

Approach 1. Integer numbers prime factorization:

Prime Factorization of a number: finding the prime numbers that multiply together to make that number.


240 = 24 × 3 × 5;
240 is not a prime, is a composite number;


160 = 25 × 5;
160 is not a prime, is a composite number;


* Positive integers that are only dividing by themselves and 1 are called prime numbers. A prime number has only two factors: 1 and itself.
* A composite number is a positive integer that has at least one factor (divisor) other than 1 and itself.


Calculate the greatest (highest) common factor (divisor), gcf, hcf, gcd:

Multiply all the common prime factors, by the lowest exponents (if any).


gcf, hcf, gcd (240; 160) = 24 × 5



gcf, hcf, gcd (240; 160) = 24 × 5 = 80;
The numbers have common prime factors.


Approach 2. Euclid's algorithm:

This algorithm involves the operation of dividing and calculating remainders.


'a' and 'b' are the two positive integers, 'a' >= 'b'.


Divide 'a' by 'b' and get the remainder, 'r'.


If 'r' = 0, STOP. 'b' = the GCF (HCF, GCD) of 'a' and 'b'.


Else: Replace ('a' by 'b') & ('b' by 'r'). Return to the division step above.



Step 1. Divide the larger number by the smaller one:
240 ÷ 160 = 1 + 80;
Step 2. Divide the smaller number by the above operation's remainder:
160 ÷ 80 = 2 + 0;
At this step, the remainder is zero, so we stop:
80 is the number we were looking for, the last remainder that is not zero.
This is the greatest common factor (divisor).


Greatest (highest) common factor (divisor):
gcf, hcf, gcd (240; 160) = 80

Why is the answer a factor (a divisor) of the initial 'a' and 'b'?

Note: 'a' ÷ 'b' = 'q' + 'r' is equivalent to the equation: 'a' = 'q' × 'b' + 'r', where 'q' is the quotient of the operation.


When the final value of 'r' = 0, the final value of 'b' is a factor (a divisor) of the final value of 'a', since 'a' = 'q' × 'b' + 0.


Go backwards the previous division steps, through each equation, 'a' = 'q' × 'b' + 'r', and notice that at each step the final value of 'b' is a factor (a divisor) of each value of 'r' and of each value of 'b' and therefore is a factor of each value of 'a'. So the final value of 'b', which is the last remainder in our list that is not zero, is a factor of the initial values of ('a' and 'b'), or in other words, is a divisor of the intial values of ('a' and 'b').


Why is the answer equal to the CGF (HCF, GCD)?

Look at all the equations: 'a' = 'q' × 'b' + 'r'. As we saw above, the final value of 'b' is a factor of all the values of 'a', 'b', and 'r'.


Therefore the final value of 'b' must also be a factor of the last value of 'r', the one that is not zero. And the final value of 'b' couldn't be larger than that value. But the final value of 'b' is actually equal to that value of 'r', therefore the final value of 'b' is the largest factor (divisor) of the initial values of 'a' and 'b'. And by definition it's called the greatest (highest) common factor (divisor) of numbers.


gcf, hcf, gcd (240; 160) = 80 = 24 × 5;

Final answer:
Greatest (highest) common factor (divisor)
gcf, hcf, gcd (240; 160) = 80 = 24 × 5;
The numbers have common prime factors.

Why do we need the greatest (highest) common factor (divisor)?

When you have calculated the greatest (highest) common factor (divisor), GCF (HCF, GCD), of the numerator and denominator of a fraction, it becomes easier to reduce it (simplify it) to the lowest terms.



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Tutoring: what is it and how to calculate the greatest common factor GCF of integers numbers (also called greatest common divisor GCD, or highest common factor, HCF)

If "t" is a factor (divisor) of "a" then among the prime factors of the prime factorization of "t" will appear only prime factors that also appear in the prime factorization of "a", and the maximum of their exponents is at most equal to those involved in the prime factorization of "a".

For example, 12 is a divisor of 60:

  • 12 = 2 × 2 × 3 = 22 × 3
  • 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5

If "t" is a common factor of "a" and "b", then the prime factorization of "t" contains only prime factors involved in the prime factorizations of both "a" and "b", by the lower powers (exponents).

For example, 12 is the common factor of 48 and 360.

  • 12 = 22 × 3
  • 48 = 24 × 3
  • 360 = 23 × 32 × 5
  • Please note that 48 and 360 have more factors (divisors): 2, 3, 4, 6, 8, 12, 24. Among them, 24 is the greatest common factor, GCF (or the greatest common divisor, GCD, or the highest common factor, HCF) of 48 and 360.

The greatest common factor, GCF, is the product of all the prime factors involved in both the prime factorizations of "a" and "b", by the lowest powers.

Based on this rule it is calculated the greatest common factor, GCF, (or greatest common divisor GCD, HCF) of several numbers, as shown in the example below:

  • 1,260 = 22 × 32
  • 3,024 = 24 × 32 × 7
  • 5,544 = 23 × 32 × 7 × 11
  • Common prime factors are: 2, its lowest power is min. (2; 3; 4) = 2; 3, its lowest power is min. (2; 2; 2) = 2;
  • GCF (1,260; 3,024; 5,544) = 22 × 32 = 252

If two numbers "a" and "b" have no other common factors (denominators) than one, gfc, gcd, hcf (a; b) = 1, then the numbers "a" and "b" are called COPRIME, or prime to each other.

If "a" and "b" are not coprime, then every common factor of "a" and "b" is a also a factor (divisor) of the greatest common factor, GCF (greatest common divisor, GCD, highest common factor, HCF) of "a" and "b".


What is a prime number?

What is a composite number?

Prime numbers up to 1,000

Prime numbers up to 10,000

Sieve of Eratosthenes

Euclid's algorithm

Simplifying ordinary (common) math fractions (reducing to lower terms): steps to follow and examples