Question 2

First, we determine the prime factorization of each number. For $26$:

$$26 = 2 \times 13$$

For $91$:

$$91 = 7 \times 13$$

The HCF (or GCD) is the product of the lowest power of common prime factors. The common prime factor between $26$ and $91$ is $13$.

$$\text{HCF}(26, 91) = 13$$

The LCM is the product of the highest power of all prime factors involved. The prime factors are $2, 7, 13$.

$$\text{LCM}(26, 91) = 2 \times 7 \times 13 = 182$$

We verify that $\text{LCM} \times \text{HCF} = \text{Product of the two numbers}$. Left Hand Side (LHS):

$$\text{LCM} \times \text{HCF} = 182 \times 13 = 2366$$

Right Hand Side (RHS):

$$26 \times 91 = 2366$$

Since LHS = RHS, the property is verified.

First, we determine the prime factorization of each number. For $510$:

$$510 = 10 \times 51 = 2 \times 5 \times 3 \times 17 = 2 \times 3 \times 5 \times 17$$

For $92$:

$$92 = 4 \times 23 = 2^2 \times 23$$

The HCF is the product of the lowest power of common prime factors. The only common prime factor is $2$. The lowest power is $2^1$.

$$\text{HCF}(510, 92) = 2$$

The LCM is the product of the highest power of all prime factors involved. The prime factors are $2, 3, 5, 17, 23$. Highest powers: $2^2, 3^1, 5^1, 17^1, 23^1$.

$$\text{LCM}(510, 92) = 2^2 \times 3 \times 5 \times 17 \times 23$$

$$\text{LCM}(510, 92) = 4 \times 3 \times 5 \times 17 \times 23 = 60 \times 391 = 23460$$

We verify that $\text{LCM} \times \text{HCF} = \text{Product of the two numbers}$. Left Hand Side (LHS):

$$\text{LCM} \times \text{HCF} = 23460 \times 2 = 46920$$

Right Hand Side (RHS):

$$510 \times 92 = 46920$$

Since LHS = RHS, the property is verified.

First, we determine the prime factorization of each number. For $336$:

$$336 = 2 \times 168 = 2^2 \times 84 = 2^3 \times 42 = 2^4 \times 21 = 2^4 \times 3 \times 7$$

For $54$:

$$54 = 2 \times 27 = 2 \times 3^3$$

The HCF is the product of the lowest power of common prime factors. The common prime factors are $2$ and $3$. Lowest power of $2$ is $2^1$. Lowest power of $3$ is $3^1$.

$$\text{HCF}(336, 54) = 2^1 \times 3^1 = 6$$

The LCM is the product of the highest power of all prime factors involved. The prime factors are $2, 3, 7$. Highest powers: $2^4, 3^3, 7^1$.

$$\text{LCM}(336, 54) = 2^4 \times 3^3 \times 7$$

$$\text{LCM}(336, 54) = 16 \times 27 \times 7 = 432 \times 7 = 3024$$

We verify that $\text{LCM} \times \text{HCF} = \text{Product of the two numbers}$. Left Hand Side (LHS):

$$\text{LCM} \times \text{HCF} = 3024 \times 6 = 18144$$

Right Hand Side (RHS):

$$336 \times 54 = 18144$$

Since LHS = RHS, the property is verified.