Highest Common Factor (HCF) of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. In contrast, Lowest Common Multiple (LCM) is the smallest positive integer that is a multiple of all the given numbers. Therefore, for any two positive integers, the HCF cannot exceed the numbers, while the LCM is at least as large as the numbers.

Highest Common Factor (HCF) of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. In contrast, Lowest Common Multiple (LCM) is the smallest positive integer that is a multiple of all the given numbers. Therefore, for any two positive integers, the HCF cannot exceed the numbers, while the LCM is at least as large as the numbers.

For any two positive integers a and b, the product of their HCF and LCM is always equal to the product of the two numbers. Notice that HCF is always a divisor of LCM. However, this product formula applies strictly to two numbers and does not hold in the same form for three or more numbers.

For any two positive integers a and b, the product of their HCF and LCM is always equal to the product of the two numbers. Notice that HCF is always a divisor of LCM. However, this product formula applies strictly to two numbers and does not hold in the same form for three or more numbers.

difference of lcm and hcf

Answer: HCF (Highest Common Factor) is the product of the smallest powers of common prime factors and divides each given number. LCM (Lowest Common Multiple) is the product of the greatest powers of all prime factors and is divisible by each given number. For two numbers $a$ and $b$ , $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$ , and $\text{HCF}$ always divides $\text{LCM}$ .

Step-by-step solution

Step 1: Define HCF and LCM conceptually

Highest Common Factor ( $\text{HCF}$ ) of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. In contrast, Lowest Common Multiple ( $\text{LCM}$ ) is the smallest positive integer that is a multiple of all the given numbers. Therefore, for any two positive integers, the $\text{HCF}$ cannot exceed the numbers, while the $\text{LCM}$ is at least as large as the numbers.

Step 2: Compare via prime factorisation method

Using the prime factorisation method, we express each number as a product of prime powers. To find the $\text{HCF}$ , we take the product of the smallest power of each common prime factor involved in the numbers. To find the $\text{LCM}$ , we take the product of the greatest power of each prime factor involved in the numbers.

Step 3: Illustrate with a concrete example

Consider the numbers $12$ and $18$ . Their prime factorisations are $12 = 2^2 \times 3$ and $18 = 2 \times 3^2$ . The common prime factors are $2$ and $3$ , with smallest powers $2^1$ and $3^1$ , giving $\text{HCF}(12, 18) = 6$ . The prime factors involved are $2$ and $3$ , with highest powers $2^2$ and $3^2$ , giving $\text{LCM}(12, 18) = 36$ .

Step 4: State the relationship between HCF and LCM

For any two positive integers $a$ and $b$ , the product of their $\text{HCF}$ and $\text{LCM}$ is always equal to the product of the two numbers. Notice that $\text{HCF}$ is always a divisor of $\text{LCM}$ . However, this product formula applies strictly to two numbers and does not hold in the same form for three or more numbers.

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