Question

For each of the following pairs of numbers, verify that the product of numbers is equal to the product of HCF and LCM.
1) 10, 15
2) 35, 40
3) 32, 48

Answer 1

Hint:To find which pair share the same product value of both the numbers and HCF, LCM; we first find the value of HCF and LCM from the numbers/pair given by factorization and then we find the product of the two along with the product of the number and then we equate both the numbers together to see if they are equal or not.

Complete step by step solution:
As given in the question, the first option is \[10,\text{ }15\]. Solving the product of the number first we can say that the product of the number is given as:
\[\Rightarrow 10\times 15=150\] …(1)
Now we move to find the HCF or highest common factor of the numbers i.e. \[10,\text{ }15\] by seeing the number we can tell that both are multiple of \[5\] and \[5\] is the greatest factor making the value
\[5\] as the HCF of the first option as show below:
\[\begin{align}
& \left. 5 \right|10,15 \\
& \left. {} \right|02,03 \\
\end{align}\]
Now we move onto the LCM and to find the LCM we again do the factorization as done above but will not stop till highest factor but will multiply all the factors:
\[\begin{align}
& \left. 5 \right|10,15 \\
& \left. 2 \right|02,03 \\
& \left. 3 \right|01,01 \\
\end{align}\]
Hence, the LCM of the numbers\[10,\text{ }15\] is \[5\times 3\times 2=30\]
Therefore, the values of the numbers \[10,\text{ }15\] form HCF of \[5\] and LCM of \[30\] and their product is \[30\times 5=150\] …(2)
As we can see that both (1) and (2) are the same value of \[150\], hence both the product of the numbers and product of HCF and LCM are the same.
As given in the question, the second option is \[35,40\]. Solving the product of the number first we can say that the product of the number is given as:
\[\Rightarrow 35\times 40=1400\] …(3)
Now we move to find the HCF or highest common factor of the numbers i.e. \[35,40\] by seeing the number we can tell that both are multiple of \[5\] and \[5\] is the greatest factor making the value \[5\] as the HCF of the first option as show below:
\[\begin{align}
& \left. 5 \right|35,40 \\
& \left. {} \right|07,08 \\
\end{align}\]
Now we move onto the LCM and to find the LCM we again do the factorization as done above but will not stop till highest factor but will multiply all the factors:
\[\begin{align}
& \left. 5 \right|35,40 \\
& \left. 7 \right|07,08 \\
& \left. 8 \right|01,08 \\
& \left. {} \right|01,01 \\
\end{align}\]
Hence, the LCM of the numbers \[35,40\] is \[5\times 7\times 8=280\]
Therefore, the values of the numbers \[35,40\] form HCF of \[5\] and LCM of \[280\] and their product is
\[280\times 5=1400\] …(4)
As we can see that both (3) and (4) are the same value of \[1400\], hence both the product of the numbers and product of HCF and LCM are the same.
As given in the question, the second option is \[32,48\]. Solving the product of the number first we can say that the product of the number is given as:
\[\Rightarrow 35\times 40=1400\] …(5)
Now we move to find the HCF or highest common factor of the numbers i.e. \[32,48\] by seeing the number we can tell that both are multiple of \[16\] and \[16\] is the greatest factor making the value
\[16\] as the HCF of the first option as show below:
\[\begin{align}
& \left. 16 \right|32,48 \\
& \left. \text{ } \right|02,03 \\
\end{align}\]
Now we move onto the LCM and to find the LCM we again do the factorization as done above but will not stop till highest factor but will multiply all the factors:
\[\begin{align}
& \left. 2 \right|32,48 \\
& \left. 2 \right|16,24 \\
& \left. 2 \right|08,12 \\
& \left. 2 \right|04,06 \\
& \left. 2 \right|02,03 \\
& \left. 3 \right|01,03 \\
& \left. 1 \right|01,01 \\
\end{align}\]
Hence, the LCM of the numbers \[32,48\] is \[2\times 2\times 2\times 2\times 2\times 3=96\]
Therefore, the values of the numbers \[32,48\] form HCF of \[16\] and LCM of \[96\] and their product is
\[16\times 96=1536\] …(6)
As we can see that both (5) and (6) are the same value of \[1536\], hence both the product of the numbers and product of HCF and LCM are the same.

Note:
The formula for the product of two numbers along with the HCF and LCM is given as:
\[\text{HCF}\times \text{LCM}=a\times b\] with \[a,b\] are the two numbers