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

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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 \(54\)

 Video Solution
Real Numbers
Ex 1.2 | Question 2

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 

  
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