Finding the LCM of 14, 21, and 35

The Least Common Multiple (LCM) of a set of numbers is the smallest positive integer that is a multiple of all the numbers in the set. To find the LCM of 14, 21, and 35, we can use the prime factorization method.

Here are the steps involved:

1. Find the prime factorization of each number.
2. Identify all prime factors that appear in any of the factorizations.
3. For each prime factor, take the highest power of that factor that appears in any of the factorizations.
4. Multiply these highest powers together to get the LCM.

Step 1: Prime Factorization of 14, 21, and 35

Let's break down each number into its prime factors:

• For 14: $14 = 2 \times 7$
• For 21: $21 = 3 \times 7$
• For 35: $35 = 5 \times 7$

We can list the prime factors for each number:

Step 2 & 3: Identify all Prime Factors and their Highest Powers

The prime factors that appear in the factorizations of 14, 21, and 35 are 2, 3, 5, and 7.

Now, let's find the highest power for each of these prime factors across the numbers:

• The highest power of 2 is $2^1$ (from 14).
• The highest power of 3 is $3^1$ (from 21).
• The highest power of 5 is $5^1$ (from 35).
• The highest power of 7 is $7^1$ (from 14, 21, and 35).

Step 4: Calculate the LCM

To find the LCM, we multiply these highest powers together:

$\text{LCM}(14, 21, 35) = 2^1 \times 3^1 \times 5^1 \times 7^1$

$\text{LCM}(14, 21, 35) = 2 \times 3 \times 5 \times 7$

$\text{LCM}(14, 21, 35) = 6 \times 5 \times 7$

$\text{LCM}(14, 21, 35) = 30 \times 7$

$\text{LCM}(14, 21, 35) = 210$

Thus, the Least Common Multiple of 14, 21, and 35 is 210.

Revision Table: LCM Calculation

Additional Information: Understanding LCM and HCF

LCM is closely related to the Highest Common Factor (HCF) or Greatest Common Divisor (GCD).

• LCM (Least Common Multiple): The smallest positive integer that is a multiple of two or more numbers. It is useful when dealing with problems involving events that repeat at regular intervals and you want to find when they will occur together again.
• HCF (Highest Common Factor): The largest positive integer that divides two or more numbers without leaving a remainder. It is useful for simplifying fractions or dividing objects into the largest possible equal groups.

There is a relationship between LCM and HCF for two positive integers, say 'a' and 'b':

$\text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b$

This relationship holds true for two numbers, but a slightly different approach is needed for three or more numbers.