Question

A school library has 280 science books and 300 math books. Students were told to stack these journals in such a way that each stack contains an equal number of books. Find the number of stacks of science and math books.

Answer 1

Hint: It is given that each stack contains an equal number of science and math books so to find the number of journals in each stack we have to take H.C.F of 280 and 300. H.C.F is calculated by writing the prime factorization of 280 and 300 and then the number of factors which are common in both 280 and 300 is the H.C.F. Now, after finding the H.C.F of 280 and 300 we will get the number of journals in each stack so to know the number of stack of science and math books divide 280 by H.C.F and 300 by H.C.F respectively.

Complete step-by-step answer:
It is given that the number of science and math books in a library is 280 and 300 respectively.
It is also given that stacking of journals is done in such a way that each stack contains an equal number of science and math journals.
As each stack contains an equal number of science and math journals so the number of journals in each stack is calculated by taking the H.C.F of 280 and 300.
H.C.F is the highest common factor. To find the H.C.F of 280 and 300, first of all write the prime factorization of 280 and 300 independently.
Prime factorization of 280 is equal to:
 $ 2\times 2\times 2\times 5\times 7 $
Prime factorization of 300 is equal to:
 $ 2\times 2\times 3\times 5\times 5 $
H.C.F of two numbers is the common factor which lies in both the numbers so H.C.F of 280 and 300 is $ 4\times 5 $ which on multiplication gives the value 20.
Hence, H.C.F of 280 and 300 is 20.
Now, each stack contains 20 journals so dividing 280 by 20 will give the number of stacks for science books.
 $ \begin{align}
  & \dfrac{280}{20} \\
 & =14 \\
\end{align} $
There are 14 stacks for science books.
To find the number of stacks for math books we are going to divide 300 by 20.
 $ \begin{align}
  & \dfrac{300}{20} \\
 & =15 \\
\end{align} $
There are 15 stacks for math books.
Hence, there are 14 and 15 stacks for science and math books respectively.

Note: Usually, it has observed that there is confusion between H.C.F and L.C.M. In the below, we are showing a trick to eliminate this confusion.
Let us assume we have to find H.C.F and L.C.M of 280 and 300.
In both the cases, first of all find the prime factorization of two numbers.
Prime factorization of 280 and 300 is shown below:
 $ \begin{align}
  & 280=\underline{2\times 2}\times 2\times \underline{5}\times 7 \\
 & 300=\underline{2\times 2}\times 3\times \underline{5}\times 5 \\
\end{align} $
In finding H.C.F, you have to write the factors which are common in both the numbers as you can see that only underlined numbers are common in both the prime factorization of 280 and 300 so the H.C.F is $ 2\times 2\times 5 $ .
In finding the L.C.M, first of all write the factors which are common in both 280 and 300 which is the H.C.F $ \left( 2\times 2\times 5=20 \right) $ then multiply this number with the remaining factors which are not common in 280 and 300. The uncommon factors are $ \left( 2\times 7\times 3\times 5=210 \right) $ , multiplying these factors with H.C.F will give $ 20\times 210 $ .