
Find the pairs of natural numbers whose least common multiple is 78 and the greatest common divisor is 13.
Answer
409.5k+ 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.
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
Out of 30 students in a class 6 like football 12 like class 7 maths CBSE

Master Class 11 Accountancy: Engaging Questions & Answers for Success

Express the following as a fraction and simplify a class 7 maths CBSE

The length and width of a rectangle are in ratio of class 7 maths CBSE

The ratio of the income to the expenditure of a family class 7 maths CBSE

How do you write 025 million in scientific notatio class 7 maths CBSE

Trending doubts
Give 10 examples for herbs , shrubs , climbers , creepers

Number of Prime between 1 to 100 is class 6 maths CBSE

How many time zones are in China class 6 social science CBSE

What are the main physical divisions of India class 6 social science CBSE

Write short notes on the island groups of India class 6 social science CBSE

Find the pairs of natural numbers whose least common class 6 maths CBSE
