GCF of 72 and 18 is the largest possible number that divides 72 and 18 exactly without any remainder. The factors of 72 and 18 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 and 1, 2, 3, 6, 9, 18 respectively. There are 3 commonly used methods to find the GCF of 72 and 18 - long division, prime factorization, and Euclidean algorithm.
What is GCF of 72 and 18?
Answer: GCF of 72 and 18 is 18.
Explanation:
The GCF of two non-zero integers, x(72) and y(18), is the greatest positive integer m(18) that divides both x(72) and y(18) without any remainder.
Methods to Find GCF of 72 and 18
Let's look at the different methods for finding the GCF of 72 and 18.
- Prime Factorization Method
- Long Division Method
- Listing Common Factors
GCF of 72 and 18 by Prime Factorization
Prime factorization of 72 and 18 is (2 × 2 × 2 × 3 × 3) and (2 × 3 × 3) respectively. As visible, 72 and 18 have common prime factors. Hence, the GCF of 72 and 18 is 2 × 3 × 3 = 18.
GCF of 72 and 18 by Long Division
GCF of 72 and 18 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.
- Step 1: Divide 72 (larger number) by 18 (smaller number).
- Step 2: Since the remainder = 0, the divisor (18) is the GCF of 72 and 18.
The corresponding divisor (18) is the GCF of 72 and 18.
GCF of 72 and 18 by Listing Common Factors
- Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
- Factors of 18: 1, 2, 3, 6, 9, 18
There are 6 common factors of 72 and 18, that are 1, 2, 3, 6, 9, and 18. Therefore, the greatest common factor of 72 and 18 is 18.
☛ Also Check:
GCF of 72 and 18 Examples
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Example 1: The product of two numbers is 1296. If their GCF is 18, what is their LCM?
Solution:
Given: GCF = 18 and product of numbers = 1296 ∵ LCM × GCF = product of numbers ⇒ LCM = Product/GCF = 1296/18
Therefore, the LCM is 72.
Example 2: Find the greatest number that divides 72 and 18 exactly.
Solution:
The greatest number that divides 72 and 18 exactly is their greatest common factor, i.e. GCF of 72 and 18.
⇒ Factors of 72 and 18:
- Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
- Factors of 18 = 1, 2, 3, 6, 9, 18
Therefore, the GCF of 72 and 18 is 18.
Example 3: For two numbers, GCF = 18 and LCM = 72. If one number is 72, find the other number.
Solution:
Given: GCF (z, 72) = 18 and LCM (z, 72) = 72 ∵ GCF × LCM = 72 × (z) ⇒ z = (GCF × LCM)/72 ⇒ z = (18 × 72)/72 ⇒ z = 18
Therefore, the other number is 18.
Show Solution >
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The GCF of 72 and 18 is 18. To calculate the greatest common factor (GCF) of 72 and 18, we need to factor each number (factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72; factors of 18 = 1, 2, 3, 6, 9, 18) and choose the greatest factor that exactly divides both 72 and 18, i.e., 18.
What is the Relation Between LCM and GCF of 72, 18?
The following equation can be used to express the relation between Least Common Multiple and GCF of 72 and 18, i.e. GCF × LCM = 72 × 18.
What are the Methods to Find GCF of 72 and 18?
There are three commonly used methods to find the GCF of 72 and 18.
- By Prime Factorization
- By Long Division
- By Euclidean Algorithm
How to Find the GCF of 72 and 18 by Prime Factorization?
To find the GCF of 72 and 18, we will find the prime factorization of the given numbers, i.e. 72 = 2 × 2 × 2 × 3 × 3; 18 = 2 × 3 × 3. ⇒ Since 2, 3, 3 are common terms in the prime factorization of 72 and 18. Hence, GCF(72, 18) = 2 × 3 × 3 = 18
☛ Prime Numbers
If the GCF of 18 and 72 is 18, Find its LCM.
GCF(18, 72) × LCM(18, 72) = 18 × 72 Since the GCF of 18 and 72 = 18 ⇒ 18 × LCM(18, 72) = 1296 Therefore, LCM = 72
☛ Greatest Common Factor Calculator
How to Find the GCF of 72 and 18 by Long Division Method?
To find the GCF of 72, 18 using long division method, 72 is divided by 18. The corresponding divisor (18) when remainder equals 0 is taken as GCF.
Given:
Product of L.C.M and H.C.F = 72
Difference of two numbers = 6
Formula used:
L.C.M × H.C.F = Product of numbers
Calculation:
Let the numbers be a and b
Now, ab = 72 ----(1)
And, a - b = 6 ----(2)
Factors of ab are
(2, 36), (3, 24), (4, 18), (6, 12), (8, 9)
As, the difference is of 6.
So, numbers are 6 and 12.
∴ The two number are 6 and 12.
Important Points
In arithmetic and number theory, the least common multiple, the lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b.
The greatest number which divides each of the two or more numbers is called HCF or Highest Common Factor.
It is also called the Greatest Common Measure(GCM) and Greatest Common Divisor(GCD).
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If HCF of two numbers is 6 and lcm of two same two numbers is 72 . Find the numbere
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$72=2^3 \cdot 3^2$
So the two numbers must look like $x = 2^a \cdot 3^b$ and $y = 2^c \cdot 3^b$.
Since $\gcd(x,y)=6=2^1 \cdot 3^1$, then $$\text{$\min(a,c)=1$ and $\min(b,d)=1$}$$.
Since $\operatorname{lcm}(x,y)=72=2^3 \cdot 3^2$, then $$\text{$\max(a,c)=3$ and $\max(b,d)=2$}$$
$$ \left \{ \begin{array}{c} \min(a,c)=1 \\ \max(a,c)=3 \end{array} \right \} \implies \{a,c\}=\{1,3\} \implies \{2^a, 2^c\}=\{2,8\} $$
$$ \left \{ \begin{array}{c} \min(b,d)=1 \\ \max(b,d)=2 \end{array} \right \} \implies \{b,d\}=\{1,2\} \implies \{3^b, 3^d\} = \{3,9\} $$
$$(x,y) \in \{ (6,72),(18,24),(24,18),(72,6) \}$$