Question 9

We will explore the Least Common Multiple (LCM) for different types of number pairs.

Step 1 — Multiples of 3 Examples

Let us take two multiples of 3. Consider the numbers 6 and 9. We list multiples of 6. Multiples of 6 are: 6, 12, 18, 24, ... We list multiples of 9. Multiples of 9 are: 9, 18, 27, ... The smallest common multiple is 18. So, LCM(6, 9) is 18.

Let us take another pair. Consider the numbers 9 and 12. We list multiples of 9. Multiples of 9 are: 9, 18, 27, 36, ... We list multiples of 12. Multiples of 12 are: 12, 24, 36, ... The smallest common multiple is 36. So, LCM(9, 12) is 36.

$\boxed{\text{LCM(6, 9) = 18, LCM(9, 12) = 36}}$

Step 2 — General Statement for Multiples of 3

We observe a pattern. Both 18 and 36 are multiples of 3. This means they can be divided by 3. The LCM of two multiples of 3 is always a multiple of 3. Let us understand why this happens. Both numbers are divisible by 3. So, 3 is a common factor for them. The LCM must contain all prime factors. It must include the factor of 3. Thus, the LCM must also be divisible by 3.

$\boxed{\text{General Statement: The LCM of two multiples of 3 is always a multiple of 3.}}$

Step 3 — Consecutive Even Numbers Examples

Let us take two consecutive even numbers. Consider the numbers 2 and 4. Multiples of 2 are: 2, 4, 6, ... Multiples of 4 are: 4, 8, ... The smallest common multiple is 4. So, LCM(2, 4) is 4.

Let us take another pair. Consider the numbers 6 and 8. Multiples of 6 are: 6, 12, 18, 24, ... Multiples of 8 are: 8, 16, 24, ... The smallest common multiple is 24. So, LCM(6, 8) is 24.

Let us take one more pair. Consider the numbers 10 and 12. Multiples of 10 are: 10, 20, 30, 40, 50, 60, ... Multiples of 12 are: 12, 24, 36, 48, 60, ... The smallest common multiple is 60. So, LCM(10, 12) is 60.

$\boxed{\text{LCM(2, 4) = 4, LCM(6, 8) = 24, LCM(10, 12) = 60}}$

Step 4 — General Statement for Consecutive Even Numbers

We observe a pattern in the examples. For (2, 4), the product is $2 \times 4 = 8$. The LCM is 4. This is half of 8. For (6, 8), the product is $6 \times 8 = 48$. The LCM is 24. This is half of 48. For (10, 12), the product is $10 \times 12 = 120$. The LCM is 60. This is half of 120. The LCM of two consecutive even numbers is half their product.

Let us understand why this happens. Let the first even number be $2n$. Let the next even number be $2n+2$. We know that LCM times HCF equals product. $\text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)}$ We need to find the HCF of $2n$ and $2n+2$. $2n$ can be written as $2 \times n$. $2n+2$ can be written as $2 \times (n+1)$. The numbers $n$ and $n+1$ are consecutive. Consecutive numbers have HCF of 1. So, the common factor for $2n$ and $2n+2$ is 2. The HCF of $2n$ and $2n+2$ is 2. Now we can find the LCM using the formula. $\text{LCM}(2n, 2n+2) = \frac{2n \times (2n+2)}{2}$ We can simplify this expression. $= n \times (2n+2)$ $= 2n^2 + 2n$ This formula gives the LCM.

$\boxed{\text{General Statement: The LCM of two consecutive even numbers } 2n \text{ and } 2n+2 \text{ is } 2n^2 + 2n.}$

Step 5 — Consecutive Numbers Examples

Let us take two consecutive numbers. Consider the numbers 7 and 8. Multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56, ... Multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, ... The smallest common multiple is 56. So, LCM(7, 8) is 56.

Let us take another pair. Consider the numbers 9 and 10. Multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, ... Multiples of 10 are: 10, 20, 30, 40, 50, 60, 70, 80, 90, ... The smallest common multiple is 90. So, LCM(9, 10) is 90.

$\boxed{\text{LCM(7, 8) = 56, LCM(9, 10) = 90}}$

Step 6 — General Statement for Consecutive Numbers

We observe a pattern in the examples. For (7, 8), the product is $7 \times 8 = 56$. The LCM is 56. This is equal to the product. For (9, 10), the product is $9 \times 10 = 90$. The LCM is 90. This is equal to the product. The LCM of two consecutive numbers is equal to their product.

Let us understand why this happens. Consecutive numbers follow each other. They never share any common factors other than 1. This means their HCF (Highest Common Factor) is always 1. Numbers with HCF of 1 are called co-prime numbers. We know that LCM times HCF equals product. $\text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)}$ Since HCF is 1 for consecutive numbers. $\text{LCM}(a, b) = \frac{a \times b}{1}$ $= a \times b$ So, the LCM is their product.

$\boxed{\text{General Statement: The LCM of two consecutive numbers is equal to their product.}}$

Step 7 — Co-prime Numbers Examples

Let us take two co-prime numbers. Co-prime numbers have only 1 as a common factor. Consider the numbers 4 and 9. Factors of 4 are: 1, 2, 4. Factors of 9 are: 1, 3, 9. Their only common factor is 1. So they are co-prime. Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, ... Multiples of 9 are: 9, 18, 27, 36, ... The smallest common multiple is 36. So, LCM(4, 9) is 36.

Let us take another pair. Consider the numbers 5 and 8. Factors of 5 are: 1, 5. Factors of 8 are: 1, 2, 4, 8. Their only common factor is 1. So they are co-prime. Multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, ... Multiples of 8 are: 8, 16, 24, 32, 40, ... The smallest common multiple is 40. So, LCM(5, 8) is 40.

$\boxed{\text{LCM(4, 9) = 36, LCM(5, 8) = 40}}$

Step 8 — General Statement for Co-prime Numbers

We observe a pattern in the examples. For (4, 9), the product is $4 \times 9 = 36$. The LCM is 36. This is equal to the product. For (5, 8), the product is $5 \times 8 = 40$. The LCM is 40. This is equal to the product. The LCM of two co-prime numbers is equal to their product.

Let us understand why this happens. Co-prime numbers have an HCF of 1. We know that LCM times HCF equals product. $\text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)}$ Since HCF is 1 for co-prime numbers. $\text{LCM}(a, b) = \frac{a \times b}{1}$ $= a \times b$ So, the LCM is their product.

$\boxed{\text{General Statement: The LCM of two co-prime numbers is equal to their product.}}$

Answer

(a) General Statement: The LCM of two multiples of 3 is always a multiple of 3. (b) General Statement: The LCM of two consecutive even numbers $2n$ and $2n+2$ is $2n^2 + 2n$. (c) General Statement: The LCM of two consecutive numbers is equal to their product. (d) General Statement: The LCM of two co-prime numbers is equal to their product.