##### HCF and LCM

## **How to find HCF and LCM?**

One method to find both HCF and LCM by using** Prime factorization.**

**Finding HCF using prime factorization**

**Step 1: **Find the prime factors of each given number.

**Step 2:** Identify all the common prime factors of the given numbers.

**Step 3: **Multiply the common factors.The product of these common factors is the HCF of the given numbers.

**Finding LCM using prime factorization**

**Step 1: **Find the prime factors of each given number.

**Step 2: **Underline the largest set of multiples of each factor.

**Step 3: **List these and multiply them out to find the LCM.

**Example 1**: Find the HCF of 36 and 60

We identify all the common factors that are **2, 2, 3**.

36 = **2 x 2 x 3** x 3.

60 = **2 **x** 2 **x** 3** x 5

We multiply the factors and the product is the HCF of 36 and 60.**HCF (36,60) = 2 x 2 x 3 = 12 **

Example 2: Find the LCM of 60 and 72 using the prime factorization.

Underline the largest set of multiples of each factor.

60 = 2 x 2 x 3 x 5

72 = 2 x 2 x 2 x 3 x 3

List these and multiply them out to find the LCM.**LCM (60, 72) = 2 x 2 x 2 x 3 x 3 x 5 = 360**

**How to find HCF using the Long Division Method**

**Step 1**: We divide the bigger number by smaller one.

**Step 2**: Divide smaller number in step 1 with remainder obtained in step 1.

**Step 3** : Divide divisor of second step with remainder obtained in step 2.

**Step 4** : We will continue this process till we get remainder zero and divisor obtained in end is the required HCF.

**Example 3: **Find the HCF of 18 and 48 by using the Long Division Method.

**How to find LCM of two numbers **

**Step 1: **Find the HCF of those numbers following the steps above.

**Step 2: **Divide the HCF into either number,it doesn´t matter which of them.

**Step 3: **Take the answer and multiply it by the other number.

**Step 4: **The number you get it´s the LCM of our two number.

**Example 4:** Find the LCM of 18 and 48

**Solution**: The HCF of 18 and 48 based on the solution above is 6.

We divide the HCF into either number. We divide 18 by 6 since it’s easier, 18÷6=3.

We take the answer and multiply it by the other number, 3 x 48 = 144.

The number 144 is the LCM of 18 and 48.

**Logarithmic Functions:**

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**Be Careful ! **You won´t be told to use the HCF or LCM to solve a problem, you need to recognize that word problems involving LCM will usually include repeating events.You may be asked how many items you need to “have enough“or when something will happen again at the same time.

**Problem 1: **Eve drives around a track in 15 minutes. John drives the same track in 20 minutes.If they start at the same place at the same time how many minutes will pass before they both cross the start line again.

**Solution: **We can see that to cross the start line both

at the same time again,this needs a repeated action wich in our case is driving

around the track again and again.Since we have repeated action we can use LCM.

Lets write both numbers 15 and 20 as product of prime factors by using the Tree Method.

**LCM (15,20) = 2 x 2 x 3 x 5 = 60.**

After 60 minutes they both will cross the same line again.