H.C.F & L.C.M

Highest Common Factor (H.C.F)

Definition:

The H.C.F of two or more numbers is the largest number that divides each of the numbers exactly without leaving a remainder. It is also known as the Greatest Common Divisor (GCD).

Methods to Find H.C.F:

1. Prime Factorization Method:

• Step 1: Write each number as a product of its prime factors.
• Step 2: Identify the common prime factors.
• Step 3: Multiply the common prime factors to get the H.C.F.

Example:

Find the H.C.F of 18 and 24.

• Prime factors of 18: $2\times 3^2$
• Prime factors of 24: $2^3\times 3$
• Common prime factors: $2\times 3$
• H.C.F: $2\times 3=6$

2. Division Method (Euclidean Algorithm):

• Step 1: Divide the larger number by the smaller number.
• Step 2: Take the remainder and divide the previous divisor by this remainder.
• Step 3: Repeat the process until the remainder is zero.
• Step 4: The last non-zero remainder is the H.C.F.

Example:

Find the H.C.F of 56 and 98.

1. $98÷56=1$ (remainder 42)
2. $56÷42=1$ (remainder 14)
3. $42÷14=3$ (remainder 0)
4. H.C.F: 14

3. Using the Relationship with L.C.M:

The H.C.F can also be found using the relationship between H.C.F and L.C.M: $HCF(a,b)=\frac{a\times b}{LCM(a,b)}$

Lowest Common Multiple (L.C.M)

Definition:

The L.C.M of two or more numbers is the smallest number that is a multiple of each of the numbers.

Methods to Find L.C.M:

1. Prime Factorization Method:

• Step 1: Write each number as a product of its prime factors.
• Step 2: Take the highest power of each prime factor present in the factorizations.
• Step 3: Multiply these highest powers to get the L.C.M.

Example:

Find the L.C.M of 18 and 24.

• Prime factors of 18: $2\times 3^2$
• Prime factors of 24: $2^3\times 3$
• Highest powers: $2^3\times 3^2$
• L.C.M: $8\times 9=72$

2. Division Method:

• Step 1: Write the numbers in a row.
• Step 2: Divide the numbers by a common prime factor.
• Step 3: Continue dividing the result by common prime factors until only 1s are left.
• Step 4: Multiply all the divisors used to get the L.C.M.

Example:

Find the L.C.M of 20 and 30.

1. Divide by 2: $20÷2=10$, $30÷2=15$
2. Divide by 2: $10÷2=5$, $15$ (not divisible)
3. Divide by 3: $5$, $15÷3=5$
4. Divide by 5: $5÷5=1$, $5÷5=1$
5. L.C.M: $2\times 2\times 3\times \mathrm{5 }= 60$

3. Using the Relationship with H.C.F:

The L.C.M can also be found using the relationship between H.C.F and L.C.M: $LCM(a,b)=\frac{a\times b}{HCF(a,b)}$

Practical Applications

H.C.F:

1. Simplifying Fractions: Reducing fractions to their simplest form by dividing the numerator and the denominator by their H.C.F.

   Example:

   Simplify the fraction $\frac{18}{24}$.

   • H.C.F of 18 and 24 is 6.
   • Simplified fraction: $\frac{18÷6}{24÷6}=\frac{3}{4}$.

2. Common Measures: Finding common measures or dimensions in real-life scenarios, such as cutting lengths of ribbon into equal parts.

L.C.M:

1. Synchronizing Events: Finding common periods for repeated events, such as the timing of traffic lights or scheduling meetings.

   Example:

   If two traffic lights change every 45 seconds and 60 seconds, find the time interval at which both lights will change simultaneously.

   • L.C.M of 45 and 60 is 180 seconds.
   • The lights will change simultaneously every 180 seconds.

2. Inventory Management: Determining reorder points for multiple items with different reorder cycles to synchronize ordering.

Understanding H.C.F and L.C.M is essential for solving various arithmetic problems and practical applications in fields such as business, logistics, and operations management.