# Ex.1.2 Q2 Real Numbers Solution - NCERT Maths Class 10

## Question

Find the LCM and HCF of the following pairs of integers and verify that LCM \(\times\) HCF \(=\) Product of the two numbers.

(i) \(26\) and \(91 \)

(ii) \(510\) and \(92\)

(iii) \(336 \) and \(54\)

## Text Solution

**What is known?**

Pairs of numbers.

**What is unknown?**

The LCM and HCF of the following pairs of integers and verify that LCM \(\times\) HCF \(=\) Product of the two numbers

**Reasoning:**

- To find the LCM and HCF of the given pairs of the integers, first find the prime factors of the given pairs of integers.
- Then, find the product of smallest power of each common factor in the numbers. This will be the HCF.
- Then find the product of greatest power of each prime factor in the number. This would be the LCM.
- Now, you have to verify LCM \(\times\) HCF \(=\) product of the two numbers, find the product of LCM and HCF and also the two given numbers. If LHS is equal to the RHS then it will be verified.

**Steps:**

(i) \(26\) and \(91\)

Prime factors of \(26=2\times 13\)

Prime factors of \(91=7\times 13\)

HCF of \(26\) and \(91=13 \)

LCM of \(26\) and \(91\)\[\begin{align}&=2 \times7 \times 3\\&=14 \times 17 \\&=\text{ }182 \\ \end{align}\]

Product of two numbers \[\begin{align}&=26\times 91\\ &=2366 \end{align}\]

LCM \(\times\) HCF

\[\begin{align}&= 182\,\times\,13\\ &= 2366\end{align}\]

So, product of two numbers \(=\) LCM \(\times\)HCF

(ii) \(510\) and \(92\)

Prime factors of \(510=2\times 3\times 5\times 17\)

Prime factors of \(92=2\,\times \,2\times 23\)

HCF of two numbers \(=2\)

LCM of two numbers

\[\begin{align}&=2\times 2\times 3\times 5\times 17\times 23 \\ &=23460\end{align}\]

Product of two numbers

\[\begin{align}&=510\,\times\,92 \\ &= 46920\end{align}\]

LCM \(\times\) HCF \[\begin{align}&=2\times 23460 \\&=46920\end{align}\]

Product of two numbers \(=\) LCM \(\times\) HCF

(iii) \(336 \) and \(54\)

Prime factors of \(336=2\times 2\times 2\times 2\times 3\times 7\)

Prime factors of \(54=2 \times 3 \times 3\times 3\)

HCF of two numbers \(=6\)

LCM of two numbers

\[\begin{align} &=2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 7 \\ &=2^{4} \times 3^{3} \times 7 \\ &=3024 \end{align}\]

Product of two numbers

\[\begin{align} &=336 \times 54 \\ &=18144 \end{align}\]

LCM \(\times\) HCF

\[\begin{align}&=3024 \times \underline{6} \\ &=18144\end{align}\]

Product duct of two numbers \(=\)LCM \(\times\) HCF