Introduction
In mathematics, the Highest Common Factor (HCF) and Least Common Multiple (LCM) are highly useful concepts for working with numbers. They are essential for solving problems related to divisibility, fractions, ratios, and more.
Highest Common Factor (HCF)
- The Highest Common Factor (HCF) is the greatest number that divides two or more integers exactly, without leaving a remainder.
- It is primarily used to find common factors and simplify numerical expressions.
Least Common Multiple (LCM)
- The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more integers.
- It is highly useful in solving problems involving common time intervals, synchronization, and operations with fractions.
What do you mean by Prime and Common factors?
FACTORS:
- A Factor of a number is any whole number that divides it exactly without leaving a remainder.
- A factor is any number that divides 20 exactly without leaving a remainder.
- The factors of 20 are: 1, 2, 4, 5, 10, and 20. This can be verified by their multiplication pairs:
$$1 \times 20 = 20$$$$2 \times 10 = 20$$$$4 \times 5 = 20$$
Final Answer: Therefore, all these numbers are factors of 20 because they divide it completely.
Now, let us understand Two different types of Common Factors:
- Prime Factors: Prime factors are the building blocks of a number they are the prime numbers that you multiply together to get that number.
- Common Factors: Common factors are the factors that two or more numbers share – in other words, the numbers that can divide all of them without leaving a remainder.
PRIME FACTORS
- Prime factors are the specific factors of a number that are also prime numbers. In other words, they are the foundational prime numbers that multiply together to produce the original number.
- A prime number is a number that has exactly two factors: 1 and itself (such as 2, 3, 5, 7, 11…).
- The numbers we can multiply to get 18 are:
$$2\times3\times3\times2$$
- So, the prime factors of 18 are 2 and 3
- We can also write it as:
Final Answer: $18=2\times3\times3\times2$ (only prime numbers)
COMMON FACTORS
- Common factors are the numbers that are factors of two or more numbers at the same time.
- Factors of $12=1,2,3,4,6,12$
- Factors of $16=1,2,4,8,16$
- The numbers they both have are: 1, 2, and 4
Final Answer: So, the common factors of 12 and 16 are 1, 2, and 4
How to calculate HCF and LCM of a number
HCF OF A NUMBER
- HCF stands for Highest Common Factor.
- It is the largest number that can exactly divide two or more given numbers without leaving a remainder.
How to Find HCF
The most famous method to find HCF is the Prime Factorization method.
Let us perform the Prime Factorization method:
- Prime Factors of 220:
2 220 2 110 5 55 11 11 1 $$\begin{aligned} 220 &= 2 \times 2 \times 5 \times 11 \\ &= 2^2 \times 5 \times 11 \end{aligned}$$ - Prime Factors of 180:
2 180 2 90 3 45 3 15 5 5 1 $$\begin{aligned} 180 &= 2 \times 2 \times 3 \times 3 \times 5 \\ &= 2^2 \times 3^2 \times 5 \end{aligned}$$ - Now take the common prime factors with the lowest powers:
- $2^2$ (common in both)
- $5$ (common in both)
- So,
$$HCF = 2^2 \times 5 = 4 \times 5 = 20$$
Final Answer: Hence, the HCF of 220 and 180 is 20.
LCM OF A NUMBER
- LCM stands for Least Common Multiple.
- It is the smallest number that is a multiple of two or more numbers.
How to Find LCM
- The method that is commonly used to find LCM is the Prime Factorization method.
- Let us perform the Prime Factorization method:
- First, find the prime factorization of each number:
- For 108:
2 108 2 54 3 27 3 9 3 3 1 $$\begin{aligned} 108 &= 2 \times 2 \times 3 \times 3 \times 3 \\ &= 2^2 \times 3^3 \end{aligned}$$ - For 120:
2 120 2 60 2 30 3 15 5 5 1 $$\begin{aligned} 120 &= 2 \times 2 \times 2 \times 3 \times 5 \\ &= 2^3 \times 3 \times 5 \end{aligned}$$ - To find the LCM, take all prime factors from both numbers, using the highest power of each:
- $2^3$ (from 120)
- $3^3$ (from 108)
- $5$ (from 120)
- Now multiply:
$$LCM = 2^3 \times 3^3 \times 5$$$$LCM = 8 \times 27 \times 5$$$$LCM = 216 \times 5 = 1080$$
Final Answer: Hence, the LCM of 108 and 120 is 1080.
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- This is a HCF problem, because we are dividing into equal parts.
- Find HCF of 60, 84, and 108.
$$60 = 2^{2} \times 3 \times 5$$$$84 = 2^{2} \times 3 \times 7$$$$108 = 2^{2} \times 3^{3}$$
- Now take the common factors:
$$\text{Common} = 2^{2} \times 3 = 12$$
Final Answer: Hence, the longest piece that can be used is of 12 cm.
(a) Is there a formula connecting HCF and LCM?
(b) If the HCF of two numbers is 1, what does that mean?
- (a) Yes! For any two positive integers a and b:
$$HCF(a,b)\times LCM(a,b)=a\times b$$This means the product of HCF and LCM of two numbers is equal to the product of the numbers themselves.
- (b) It means the numbers are co-prime (No common factors). In such cases:
$$HCF=1\Rightarrow LCM=a\times b$$
- Using the Prime Factorization method:
- HCF:
$$15 = 3 \times 5$$$$25 = 5 \times 5 = 5^2$$The common prime factor is $5$.$$HCF = 5$$Therefore, the HCF of 15 and 25 comes out to be 5.
- LCM:
Take all prime factors with their highest powers:
$$LCM = 3 \times 5^2$$$$LCM = 3 \times 25$$$$LCM = 75$$
Final Answer: Therefore, the LCM of 15 and 25 comes out to be 75.