Questions on LCM and HCF

(b) 12
Prime Factorization:
$24 = 2^3 \times 3$
$36 = 2^2 \times 3^2$
HCF: Lowest powers of common prime factors: $2^2 \times 3 = 12$

(a) 75
Prime Factorization:
$15 = 3 \times 5$
$25 = 5^2$
LCM: Highest powers: $3 \times 5^2 = 75$

(a) 15
Prime Factorization:
$45 = 3^2 \times 5$
$60 = 2^2 \times 3 \times 5$
HCF: Common factors: $3 \times 5 = 15$

(a) 24
Prime Factorization:
$8 = 2^3$
$12 = 2^2 \times 3$
LCM: Highest powers: $2^3 \times 3 = 24$

(b) 14
Prime Factorization:
$56 = 2^3 \times 7$
$84 = 2^2 \times 3 \times 7$
$126 = 2 \times 3^2 \times 7$
HCF: Lowest powers of common factors: $2 \times 7 = 14$

(b) 2520.

To find the LCM (Least Common Multiple) of the numbers 40, 36, and 126, we use their prime factorizations:

• • $40 = 2^3 \times 5$
    
  • $36 = 2^2 \times 3^2$
    
  • $126 = 2 \times 3^2 \times 7$
    

  To calculate the LCM, we take the highest power of each prime factor present in the factorizations:

  • The highest power of 2 is $2^3$.
  • The highest power of 3 is $3^2$.
  • The highest power of 5 is $5.$
  • The highest power of 7 is $7$.

  So, the LCM is:

  $$LCM=2^3\times 3^2\times 5\times 7=8\times 9\times 5\times 7=2520$$

  Thus, the correct answer is (b) 2520.

(b) 1680

To find the LCM (Least Common Multiple) of the numbers 112, 140, and 168, we first determine their prime factorizations:

• • $112 = 2^4 \times 7$
    
  • $140 = 2^2 \times 5 \times 7$
    
  • $168 = 2^3 \times 3 \times 7$
    

  To calculate the LCM, we take the highest power of each prime factor present in the factorizations:

  • The highest power of 2 is $2^4$.
  • The highest power of 3 is $3.$
  • The highest power of 5 is $5$.
  • The highest power of 7 is $7$.

  So, the LCM is:

  $$LCM=2^4\times 3\times 5\times 7=16\times 3\times 5\times 7=1680$$

  Thus, the correct answer is (b) 1680.

(a) 7.2.

Here’s a step-by-step breakdown of how to find the LCM of the decimal numbers 2.4, 0.36., and 7.2:

1. 1. Convert decimals to whole numbers:

      • $2.4$ becomes $240$ by multiplying by $100$.
      • $0.36$ becomes $36$ by multiplying by $100$.
      • $7.2$ becomes $720$ by multiplying by $100$.

   2. Find the LCM of these whole numbers:

      • $240 = 2^4 \times 3 \times 5$
        
      • $36 = 2^2 \times 3^2$
      • $720 = 2^4 \times 3^2 \times 5$
        

      The LCM of $240$, $36$, and $720$ is $720$.

   3. Convert back to the original scale:

      • Since we multiplied by $100$, we now divide the LCM $720$ by $100$to get back to the original decimal scale.

   Therefore, the LCM of $2.4$, $0.36$ and $7.2$is $7.2$
   

   The correct answer is (a) 7.2.

(a) 1/84.

To find the HCF (Highest Common Factor) of the fractions $\frac{3}{4}$, $\frac{5}{6}$, and $\frac{6}{7}$, we use the following formula:

$$\text{HCF of fractions} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}$$

Step 1: Find the HCF of the numerators (3, 5, 6)

• The HCF of 3, 5, and 6 is 1 because 1 is the only common factor of these numbers.

Step 2: Find the LCM of the denominators (4, 6, 7)

• Prime factorization:

  • $4 = 2^2$
    
  • $6 = 2 \times 3$
    
  • $7 = 7$
    

• The LCM of 4, 6, and 7 is:

$$LCM = 2^2 \times 3 \times 7 = 84$$

Step 3: Calculate the HCF of the fractions

$$\text{HCF}=\frac{1}{\mathrm{84<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"></span>}}$$

Thus, the correct answer is (a) 1/84.

(b) 2813.

To solve this problem, we can use the relationship between the HCF (GCD), LCM, and the product of two numbers. The formula is:

$$\text{Product of the two numbers} = \text{HCF} \times \text{LCM}$$

Given:

• LCM = 64699
• HCF = 97
• One of the numbers = 2231
• The other number = $x$

Using the formula:

$$2231\times x=97\times 64699$$

To find $x$:

$$x=\frac{97\times 64699}{\mathrm{2231<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"></span>}}$$

Calculating the value:

$$x=\frac{6275803}{2231}=2813$$

So, the other number is 2813.

Thus, the correct answer is (b) 2813.