Question

What is the prime factorization of 375 and of 1000?

Answer 1

Hint: To find the prime factorization of 375 and 1000, we are going to factorize 375 first and then 1000. It is done by dividing 375 with the lowest prime number which can be divided by 375, then whatsoever the quotient will remain, we will see by which number that remaining quotient will get divisible. In this way, we will divide till we get the quotient as 1. Similarly, we are going to do 1000.

Complete step-by-step answer:
In the above problem, we are asked to find the prime factorization of 375 and 1000. For that, first of all we are going to find the prime factorization for 375 which we are going to do by dividing 375 by a number in such a way so that 375 will get completely divided. As the last number of 57 is not an even number so 57 won’t get divided by 2 and the sum of three numbers (3, 5 and 7) is 15 and 15 is divisible by 3 so 375 will get divided by 3.
Dividing 375 by 3 we get,
\[\begin{matrix}
   3 \\
   {} \\
\end{matrix}\left| \!{\underline {\,
  \begin{align}
  & 375 \\
 & 125 \\
\end{align} \,}} \right. \]
From the above division, you can see that when we divide 357 by 3 we get 125.
Now, 125 is divisible by 5 so dividing 125 by 5 we get,
$\begin{matrix}
   3 \\
   5 \\
   {} \\
\end{matrix}\left| \!{\underline {\,
  \begin{align}
  & 375 \\
 & 125 \\
 & 25 \\
\end{align} \,}} \right. $
Again, 25 is completely divisible by 5 and we get,
$\begin{matrix}
   3 \\
   5 \\
   5 \\
   {} \\
\end{matrix}\left| \!{\underline {\,
  \begin{align}
  & 375 \\
 & 125 \\
 & 25 \\
 & 5 \\
\end{align} \,}} \right. $
Now, dividing 5 by 5 we get,
\[\begin{matrix}
   3 \\
   5 \\
   5 \\
   5 \\
   {} \\
\end{matrix}\left| \!{\underline {\,
  \begin{align}
  & 375 \\
 & 125 \\
 & 25 \\
 & 5 \\
 & 1 \\
\end{align} \,}} \right. \]
Hence, we got 1 as the quotient so we are stopping the division here and we got the prime factorization for 375 as follows:
$375=3\times 5\times 5\times 5$
Similarly, we are going to find the prime factorization for 1000 as follows:
As last number in 1000 is even so 1000 is divisible by 2 and we get,
\[\begin{matrix}
   2 \\
   2 \\
   2 \\
   {} \\
\end{matrix}\left| \!{\underline {\,
  \begin{align}
  & 1000 \\
 & 500 \\
 & 250 \\
 & 125 \\
\end{align} \,}} \right. \]
Now, the division of 125 is done in the same way in which we have done above when we were finding the prime factorization of 375.
\[\begin{matrix}
   2 \\
   2 \\
   2 \\
   5 \\
   5 \\
   5 \\
   {} \\
\end{matrix}\left| \!{\underline {\,
  \begin{align}
  & 1000 \\
 & 500 \\
 & 250 \\
 & 125 \\
 & 25 \\
 & 5 \\
 & 1 \\
\end{align} \,}} \right. \]
From the above, we have found the prime factorization for 1000 as:
$1000=2\times 2\times 2\times 5\times 5\times 5$
Hence, we have found the prime factorization for both the given numbers (i.e. 375 and 1000).

Note: The mistake that could be possible in the above problem is that you might write 1 as the prime factor which is wrong because 1 is not a prime number. You can say, 1 is a factor of 375 and 1000 but not a prime factor so make sure, you won’t make this mistake.