HCF and LCM: Relationship & Applications

The Highest Common Factor (HCF) and the Least Common Multiple (LCM) are fundamental concepts in number theory. They describe the relationships between the divisors and multiples of a set of numbers. The HCF of two or more numbers is the largest number that divides all of them exactly. The LCM of two or more numbers is the smallest number that is a multiple of all of them.

A crucial relationship exists between the HCF, LCM, and the original numbers themselves: the product of the HCF and the LCM of two numbers is always equal to the product of the two numbers.

Formulae

The core formula governing the relationship is:

$HCF(a, b) \times LCM(a, b) = a \times b$

Where:

• $a$ and $b$ are the two numbers.
• $HCF(a, b)$ is the Highest Common Factor of $a$ and $b$.
• $LCM(a, b)$ is the Least Common Multiple of $a$ and $b$.

Examples

Example-1:

Let’s consider the numbers 12 and 18.

• The factors of 12 are 1, 2, 3, 4, 6, and 12.
• The factors of 18 are 1, 2, 3, 6, 9, and 18.
• Therefore, $HCF(12, 18) = 6$.
• The multiples of 12 are 12, 24, 36, 48,…
• The multiples of 18 are 18, 36, 54, 72,…
• Therefore, $LCM(12, 18) = 36$.

Applying the formula: $HCF \times LCM = 6 \times 36 = 216$. Also, the product of the numbers $12 \times 18 = 216$. Thus the relationship holds.

Example-2:

Let’s consider the numbers 15 and 20.

• Factors of 15: 1, 3, 5, 15
• Factors of 20: 1, 2, 4, 5, 10, 20
• $HCF(15, 20) = 5$
• Multiples of 15: 15, 30, 45, 60, 75,…
• Multiples of 20: 20, 40, 60, 80,…
• $LCM(15, 20) = 60$

Applying the formula: $HCF \times LCM = 5 \times 60 = 300$. Also, the product of the numbers $15 \times 20 = 300$. Thus the relationship holds.

Theorem with Proof

Theorem: The product of the HCF and LCM of two numbers is equal to the product of the numbers.

Proof:

Let’s consider two numbers, $a$ and $b$. We can express $a$ and $b$ in terms of their prime factorizations:

$a = p_1^{a_1} \times p_2^{a_2} \times … \times p_n^{a_n}$

$b = p_1^{b_1} \times p_2^{b_2} \times … \times p_n^{b_n}$

where $p_1, p_2, …, p_n$ are prime numbers and $a_i$ and $b_i$ are non-negative integers (the exponents).

The HCF of $a$ and $b$ is the product of the lowest powers of the common prime factors:

$HCF(a, b) = p_1^{min(a_1, b_1)} \times p_2^{min(a_2, b_2)} \times … \times p_n^{min(a_n, b_n)}$

The LCM of $a$ and $b$ is the product of the highest powers of all prime factors (both common and uncommon):

$LCM(a, b) = p_1^{max(a_1, b_1)} \times p_2^{max(a_2, b_2)} \times … \times p_n^{max(a_n, b_n)}$

Now, consider the product of $HCF(a, b)$ and $LCM(a, b)$:

$HCF(a, b) \times LCM(a, b) = (p_1^{min(a_1, b_1)} \times p_2^{min(a_2, b_2)} \times … \times p_n^{min(a_n, b_n)}) \times (p_1^{max(a_1, b_1)} \times p_2^{max(a_2, b_2)} \times … \times p_n^{max(a_n, b_n)})$

For each prime factor $p_i$, the exponent in $HCF \times LCM$ is $min(a_i, b_i) + max(a_i, b_i)$. We know that for any two numbers, $x$ and $y$, $min(x, y) + max(x, y) = x + y$. Therefore:

$HCF(a, b) \times LCM(a, b) = p_1^{a_1 + b_1} \times p_2^{a_2 + b_2} \times … \times p_n^{a_n + b_n}$

This is exactly equal to the product of $a$ and $b$:

$a \times b = (p_1^{a_1} \times p_2^{a_2} \times … \times p_n^{a_n}) \times (p_1^{b_1} \times p_2^{b_2} \times … \times p_n^{b_n}) = p_1^{a_1 + b_1} \times p_2^{a_2 + b_2} \times … \times p_n^{a_n + b_n}$

Hence, $HCF(a, b) \times LCM(a, b) = a \times b$. The theorem is proven.

Common mistakes by students

Students often make these mistakes:

• Incorrect Calculation of HCF or LCM: Miscalculating the HCF or LCM leads to incorrect results. Make sure you are using the correct methods for finding HCF and LCM (prime factorization, division method, etc.).
• Confusing HCF and LCM: Mixing up the definitions of HCF (greatest common factor) and LCM (least common multiple) can lead to wrong answers.
• Applying the formula to more than two numbers directly: While the concept of HCF and LCM extends to more than two numbers, the formula $HCF \times LCM = a \times b$ is specifically for *two* numbers. Applying it directly to three or more numbers without considering the pairwise relationships is a common mistake.

Real Life Application

This relationship has practical applications in various scenarios:

• Scheduling: Determining when events (e.g., bus routes, meetings) will coincide or overlap.
• Dividing Resources: Finding the largest equal portions you can divide something into (e.g., dividing items among a group of people).
• Measurement Conversions: Solving problems involving different units of measurement.
• Optimizing Production: Calculating the minimum amount of resources needed for production and packaging.

Fun Fact

The concept of HCF and LCM is used in cryptography and computer science, especially in algorithms related to finding the greatest common divisor (GCD), which is the same as HCF. It’s a fundamental concept in making systems secure.

Recommended YouTube Videos for Deeper Understanding

Q.1 If the HCF of two numbers is 12 and their LCM is 60, and one of the numbers is 24, then the other number is:

Check Solution

Ans: A

Using the relationship, $HCF \times LCM = number1 \times number2$. We have $12 \times 60 = 24 \times number2$. Solving for $number2$, we get $number2 = (12 \times 60) / 24 = 30$.

Q.2 The product of two numbers is 180 and their HCF is 3. Find their LCM.

Check Solution

Ans: A

Using the relationship, $HCF \times LCM = product \ of \ numbers$. We have $3 \times LCM = 180$. Solving for $LCM$, we get $LCM = 180 / 3 = 60$.

Q.3 The LCM of two numbers is 90. If the numbers are 15 and x, and their HCF is 3, then find the value of x.

Check Solution

Ans: C

$HCF \times LCM = number1 \times number2$. We have $3 \times 90 = 15 \times x$. Solving for $x$, we get $x = (3 \times 90) / 15 = 18$.

Q.4 The HCF of two numbers is 5 and their LCM is 150. One number is 25. What is the other number?

Check Solution

Ans: B

$HCF \times LCM = number1 \times number2$. We have $5 \times 150 = 25 \times number2$. Solving for the other number, $number2 = (5 \times 150) / 25 = 30$.

Q.5 If the product of two numbers is 360 and their LCM is 60, then their HCF is:

Check Solution

Ans: A

Using the relationship, $HCF \times LCM = product \ of \ numbers$. We have $HCF \times 60 = 360$. Solving for $HCF$, we get $HCF = 360 / 60 = 6$.

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