Answer

Verified

367.5k+ views

**Hint:**Here, first of all, let the two numbers be x and y. Now, according to the property, the product of two natural numbers will be equal to the product of its LCM and GCD. So, therefore, we will get an equation and on solving that equation, we will get our two numbers.

**Complete step-by-step solution:**

In this question, we are given that the least common multiple (LCM) of two numbers is 78 and the greatest common divisor (GCD) is 13.

Let these two numbers be x and y.

Now, we have a property that the product of two natural numbers x and y is equal to the product of its LCM and GCD. Therefore, we can say that for two natural numbers x and y,

$ \Rightarrow x \times y = GCD\left( {x,y} \right) \times LCM\left( {x,y} \right)$

Here, we have GCQ equal to 13 and the LCM as 78. Therefore, substituting these values in above equation, we get

$\Rightarrow x \times y = 13 \times 78 \\

\Rightarrow x \times y = 1014 $

Now, on prime factorization 1014, we get

$ \Rightarrow x \times y = 2 \times 3 \times 13 \times 13$

Now, 13 is common divisor of both numbers x and y, we can write

$ \Rightarrow x \times y = \left( {2 \times 13} \right) \times \left( {3 \times 13} \right)$

OR

$ \Rightarrow x \times y = \left( {1 \times 13} \right) \times \left( {2 \times 3 \times 13} \right)$

Hence, now x will be equal to $\left( {2 \times 13} \right)$ or $\left( {1 \times 13} \right)$ and y will be equal to $\left( {3 \times 13} \right)$ or $\left( {2 \times 3 \times 13} \right)$. Therefore,

$ \Rightarrow x = 2 \times 13 = 26$ OR $ \Rightarrow x = 1 \times 13 = 13$

$ \Rightarrow y = 3 \times 13 = 39$ OR $ \Rightarrow y = 2 \times 3 \times 13 = 78$

**Hence, the two natural numbers whose LCM is 78 and GCD is 13 are 26 and 39 or 13 and 78.**

**Note:**Here, we can cross verify our answer by finding the LCM and GCD of 26 and 39.

First of all let us find LCM of 26 and 39.

$

\Rightarrow 26 = 2 \times 13 \\

\Rightarrow 39 = 3 \times 13 $

Therefore,

$ \Rightarrow LCM = 2 \times 3 \times 13 = 78$

Now, let us find the GCD of 26 and 78.

$

\Rightarrow 26 = 2 \times 13 \\

\Rightarrow 39 \Rightarrow 3 \times 13 $

Hence,

$ \Rightarrow GCD = 13$

Hence, our answer is correct.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

10 examples of evaporation in daily life with explanations

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE