HCF of 12, 15 and 18
HCF of 12, 15 and 18 is the largest possible number that divides 12, 15 and 18 exactly without any remainder. The factors of 12, 15 and 18 are (1, 2, 3, 4, 6, 12), (1, 3, 5, 15) and (1, 2, 3, 6, 9, 18) respectively. There are 3 commonly used methods to find the HCF of 12, 15 and 18  Euclidean algorithm, prime factorization, and long division.
1.  HCF of 12, 15 and 18 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is HCF of 12, 15 and 18?
Answer: HCF of 12, 15 and 18 is 3.
Explanation:
The HCF of three nonzero integers, x(12), y(15) and z(18), is the highest positive integer m(3) that divides x(12), y(15) and z(18) without any remainder.
Methods to Find HCF of 12, 15 and 18
Let's look at the different methods for finding the HCF of 12, 15 and 18.
 Prime Factorization Method
 Long Division Method
 Using Euclid's Algorithm
HCF of 12, 15 and 18 by Prime Factorization
Prime factorization of 12, 15 and 18 is (2 × 2 × 3), (3 × 5) and (2 × 3 × 3) respectively. As visible, 12, 15 and 18 have only one common prime factor i.e. 3. Hence, the HCF of 12, 15 and 18 is 3.
HCF of 12, 15 and 18 by Long Division
HCF of 12, 15 and 18 can be represented as HCF of (HCF of 12, 15) and 18. HCF(12, 15, 18) can be thus calculated by first finding HCF(12, 15) using long division and thereafter using this result with 18 to perform long division again.
 Step 1: Divide 15 (larger number) by 12 (smaller number).
 Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (12) by the remainder (3). Repeat this process until the remainder = 0.
⇒ HCF(12, 15) = 3.  Step 3: Now to find the HCF of 3 and 18, we will perform a long division on 18 and 3.
 Step 4: For remainder = 0, divisor = 3 ⇒ HCF(3, 18) = 3
Thus, HCF(12, 15, 18) = HCF(HCF(12, 15), 18) = 3.
HCF of 12, 15 and 18 by Euclidean Algorithm
As per the Euclidean Algorithm, HCF(X, Y) = HCF(Y, X mod Y)
where X > Y and mod is the modulo operator.
HCF(12, 15, 18) = HCF(HCF(12, 15), 18)
 HCF(15, 12) = HCF(12, 15 mod 12) = HCF(12, 3)
 HCF(12, 3) = HCF(3, 12 mod 3) = HCF(3, 0)
 HCF(3, 0) = 3 (∵ HCF(X, 0) = X, where X ≠ 0)
Steps for HCF(3, 18)
 HCF(18, 3) = HCF(3, 18 mod 3) = HCF(3, 0)
 HCF(3, 0) = 3 (∵ HCF(X, 0) = X, where X ≠ 0)
Therefore, the value of HCF of 12, 15 and 18 is 3.
☛ Also Check:
 HCF of 0 and 6 = 6
 HCF of 391, 425 and 527 = 17
 HCF of 7 and 8 = 1
 HCF of 84 and 98 = 14
 HCF of 90 and 120 = 30
 HCF of 12, 36 and 48 = 12
 HCF of 20 and 35 = 5
HCF of 12, 15 and 18 Examples

Example 1: Calculate the HCF of 12, 15, and 18 using LCM of the given numbers.
Solution:
Prime factorization of 12, 15 and 18 is given as,
 12 = 2 × 2 × 3
 15 = 3 × 5
 18 = 2 × 3 × 3
LCM(12, 15) = 60, LCM(15, 18) = 90, LCM(18, 12) = 36, LCM(12, 15, 18) = 180
⇒ HCF(12, 15, 18) = [(12 × 15 × 18) × LCM(12, 15, 18)]/[LCM(12, 15) × LCM (15, 18) × LCM(18, 12)]
⇒ HCF(12, 15, 18) = (3240 × 180)/(60 × 90 × 36)
⇒ HCF(12, 15, 18) = 3.
Therefore, the HCF of 12, 15 and 18 is 3. 
Example 2: Verify the relation between the LCM and HCF of 12, 15 and 18.
Solution:
The relation between the LCM and HCF of 12, 15 and 18 is given as, HCF(12, 15, 18) = [(12 × 15 × 18) × LCM(12, 15, 18)]/[LCM(12, 15) × LCM (15, 18) × LCM(12, 18)]
⇒ Prime factorization of 12, 15 and 18: 12 = 2 × 2 × 3
 15 = 3 × 5
 18 = 2 × 3 × 3
∴ LCM of (12, 15), (15, 18), (12, 18), and (12, 15, 18) is 60, 90, 36, and 180 respectively.
Now, LHS = HCF(12, 15, 18) = 3.
And, RHS = [(12 × 15 × 18) × LCM(12, 15, 18)]/[LCM(12, 15) × LCM (15, 18) × LCM(12, 18)] = [(3240) × 180]/[60 × 90 × 36]
LHS = RHS = 3.
Hence verified. 
Example 3: Find the highest number that divides 12, 15, and 18 completely.
Solution:
The highest number that divides 12, 15, and 18 exactly is their highest common factor.
 Factors of 12 = 1, 2, 3, 4, 6, 12
 Factors of 15 = 1, 3, 5, 15
 Factors of 18 = 1, 2, 3, 6, 9, 18
The HCF of 12, 15, and 18 is 3.
∴ The highest number that divides 12, 15, and 18 is 3.
FAQs on HCF of 12, 15 and 18
What is the HCF of 12, 15 and 18?
The HCF of 12, 15 and 18 is 3. To calculate the HCF (Highest Common Factor) of 12, 15 and 18, we need to factor each number (factors of 12 = 1, 2, 3, 4, 6, 12; factors of 15 = 1, 3, 5, 15; factors of 18 = 1, 2, 3, 6, 9, 18) and choose the highest factor that exactly divides 12, 15 and 18, i.e., 3.
What is the Relation Between LCM and HCF of 12, 15 and 18?
The following equation can be used to express the relation between Least Common Multiple and HCF of 12, 15 and 18, i.e. HCF(12, 15, 18) = [(12 × 15 × 18) × LCM(12, 15, 18)]/[LCM(12, 15) × LCM (15, 18) × LCM(12, 18)].
☛ Highest Common Factor Calculator
Which of the following is HCF of 12, 15 and 18? 3, 29, 62, 22, 46
HCF of 12, 15, 18 will be the number that divides 12, 15, and 18 without leaving any remainder. The only number that satisfies the given condition is 3.
What are the Methods to Find HCF of 12, 15 and 18?
There are three commonly used methods to find the HCF of 12, 15 and 18.
 By Euclidean Algorithm
 By Prime Factorization
 By Long Division
How to Find the HCF of 12, 15 and 18 by Prime Factorization?
To find the HCF of 12, 15 and 18, we will find the prime factorization of given numbers, i.e. 12 = 2 × 2 × 3; 15 = 3 × 5; 18 = 2 × 3 × 3.
⇒ Since 3 is the only common prime factor of 12, 15 and 18. Hence, HCF(12, 15, 18) = 3.
☛ Prime Number
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