HCF/LCM using Prime Factors

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What Are HCF and LCM?

HCF (Highest Common Factor): This is the largest number that can divide two or more numbers without leaving a remainder.
LCM (Lowest Common Multiple): This is the smallest number that is a multiple of two or more numbers.

What Are Prime Factors?

Before diving into HCF (Highest Common Factor) and LCM (Lowest Common Multiple), it's important to understand what prime factors are.

A prime number is a number that can only be divided by 1 and itself without leaving a remainder. Examples include 2, 3, 5, 7, 11, and so on.
A factor of a number is a whole number that divides exactly into that number.
Prime factors are factors of a number that are also prime numbers. For example, the prime factors of 12 are 2 and 3, because 12 can be divided by 2 and 3, and these numbers are prime.

How to Find Prime Factors

To break a number down into its prime factors, you can use a process called prime factorisation. This involves dividing the number by the smallest possible prime numbers until you can't divide anymore.

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Example: Finding Prime Factors of 60

Start with the smallest prime number, which is 2: $( 60 \div 2 = 30)$

$( 30 \div 2 = 15)$

Move to the next smallest prime number, which is 3: $( 15 \div 3 = 5 )$
Finally, 5 is a prime number, so we stop. Now, we can write 60 as a product of its prime factors: $[ 60 = 2 \times 2 \times 3 \times 5 ]$

Or, using exponents to show repeated factors: $[ 60 = 2^2 \times 3^1 \times 5^1]$

Finding the HCF and LCM Using Prime Factors

What Is HCF (Highest Common Factor)?

The HCF is the largest number that can divide two or more numbers without leaving a remainder.

What Is LCM (Lowest Common Multiple)?

The LCM is the smallest number that is a multiple of two or more numbers.

It's the smallest number that all the original numbers can divide into evenly.

How to Find the HCF and LCM Using Prime Factors

To find the HCF and LCM using prime factors, we follow these steps:

Prime Factorisation: Break down each number into its prime factors.
HCF: For each prime number, take the lowest power that appears in all the prime factorisations.
LCM: For each prime number, take the highest power that appears in any of the prime factorisations.
Multiply these factors together to get the HCF or LCM.

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Example: Finding the HCF and LCM of 60 and 80 Let's go through the process step by step.

Step 1: Prime Factorisation

First, we need to find the prime factors of both 60 and 80.

Prime Factors of 60:

Start by dividing 60 by the smallest prime number, which is 2:

$( 60 \div 2 = 30)$
$( 30 \div 2 = 15 )$

Next, divide by the next smallest prime number, which is 3:

$( 15 \div 3 = 5 )$

Since 5 is a prime number, we stop here. So, $( 60 = 2^2 \times 3^1 \times 5^1 ).$

Prime Factors of 80:

Start by dividing 80 by 2:

$( 80 \div 2 = 40 )$
$( 40 \div 2 = 20 )$
$( 20 \div 2 = 10 )$
$( 10 \div 2 = 5 )$

Since 5 is a prime number, we stop here. So, $( 80 = 2^4 \times 5^1 )$ .

Now we have the prime factorisations:

$( 60 = 2^2 \times 3^1 \times 5^1 )$
$( 80 = 2^4 \times 5^1 )$

Step 2: Finding the HCF (Highest Common Factor)

To find the HCF:

Look for the lowest power of each prime number that appears in both factorisations.

Prime number 2:

In 60, the power of 2 is $( 2^2 )$ .
In 80, the power of 2 is $( 2^4 )$ .
The lowest power is $( 2^2 )$ .

Prime number 3:

In 60, the power of 3 is $( 3^1 )$ .
In 80, there is no 3 in its prime factorisation.
Since 3 is not common in both numbers, we exclude it from the HCF.

Prime number 5:

Both 60 and 80 have $( 5^1 )$ .
The lowest power is $( 5^1 )$ . Now multiply these lowest powers together: $[ HCF = 2^2 \times 5^1 = 4 \times 5 = 20 ]$

Answer: The HCF of 60 and 80 is 20.

Step 3: Finding the LCM (Lowest Common Multiple)

To find the LCM:

Look for the highest power of each prime number that appears in any of the factorisations.

Prime number 2:

The highest power is $( 2^4 )$ (from 80).

Prime number 3:

The highest power is $( 3^1 )$ (from 60).

Prime number 5:

The highest power is $( 5^1 )$ (common in both 60 and 80). Now multiply these highest powers together: $[ LCM = 2^4 \times 3^1 \times 5^1 = 16 \times 3 \times 5 = 240 ]$

Answer: The LCM of 60 and 80 is 240.

Using a Calculator for Prime Factorisation

If you have a scientific calculator, you can use it to find prime factors easily:

Type in the number (like 60).
Press the "Shift" key and then the button with the factorisation symbol (usually labelled :highlight[FACT] or something similar). This will display the prime factorisation on the screen, saving you time.

Why Is This Useful?

HCF: Helps in simplifying fractions or dividing things into smaller, equal parts.
LCM: Useful in finding common time intervals, like when two events will happen together.

Recap

Prime factorisation breaks a number down into the prime numbers that multiply together to make it.
HCF: Find the lowest common powers of the prime factors.
LCM: Find the highest common powers of the prime factors.

HCF/LCM using Prime Factors Simplified Revision Notes for Junior Cycle Mathematics