Question

Express the following number as the product of prime factors:
512

Answer 1

Hint: We solve this problem by using the prime factorisation method. The prime factorisation method is simple: we divide the given number with prime numbers starting from 2. If the number is divisible then we apply the same factorisation method for the quotient. We carry on this process until we get the quotient as a prime number. Then we can write the given number as a product of all the prime numbers that we used in the division.

Complete step-by-step answer:
We are asked to express the number 512 as a product of prime numbers.
Let us assume that the given number as
\[\Rightarrow A=512\]
We know that the prime factorisation is that we divide the given number with prime numbers starting from 2
Now, let us divide the number 512 with 2 then we get
\[\Rightarrow \dfrac{512}{2}=256\]
Here we can see that the number 512 is exactly divisible by 2
So, we can write the given number as
\[\Rightarrow A=2\times 256\]
Now, we know that we need to apply the prime factorisation method to the quotient until we get the quotient as a prime number.
Now, by dividing the number 256 with 2 we get
\[\Rightarrow \dfrac{256}{2}=128\]
Here we can see that the number 256 is exactly divisible by 2
So, we can write the given number as
\[\begin{align}
  & \Rightarrow A=2\times 2\times 128 \\
 & \Rightarrow A={{2}^{2}}\times 128 \\
\end{align}\]
Now, we know that we need to apply the prime factorisation method to the quotient until we get the quotient as a prime number.
Now, by dividing the number 128 with 2 we get
\[\Rightarrow \dfrac{128}{2}=64\]
Here we can see that the number 128 is exactly divisible by 2
So, we can write the given number as
\[\begin{align}
  & \Rightarrow A={{2}^{2}}\times 2\times 64 \\
 & \Rightarrow A={{2}^{3}}\times 64 \\
\end{align}\]
Now, we know that we need to apply the prime factorisation method to the quotient until we get the quotient as a prime number.
Now, by dividing the number 64 with 2 we get
\[\Rightarrow \dfrac{64}{2}=32\]
Here we can see that the number 64 is exactly divisible by 2
So, we can write the given number as
\[\begin{align}
  & \Rightarrow A={{2}^{3}}\times 2\times 32 \\
 & \Rightarrow A={{2}^{4}}\times 32 \\
\end{align}\]
Now, we know that we need to apply the prime factorisation method to the quotient until we get the quotient as a prime number.
Now, by dividing the number 32 with 2 we get
\[\Rightarrow \dfrac{32}{2}=16\]
Here we can see that the number 32 is exactly divisible by 2
So, we can write the given number as
\[\begin{align}
  & \Rightarrow A={{2}^{4}}\times 2\times 32 \\
 & \Rightarrow A={{2}^{5}}\times 32 \\
\end{align}\]
We know that the number 32 can be represented as
\[\Rightarrow 32={{2}^{5}}\]
By substituting the value of 32 in above equation we get
\[\begin{align}
  & \Rightarrow A={{2}^{4}}\times {{2}^{5}} \\
 & \Rightarrow A={{2}^{9}} \\
\end{align}\]
Therefore the number 512 can be represented as the product of prime numbers as
\[\therefore 512={{2}^{9}}\]

So, the correct answer is “Option A”.

Note: We can have a shortcut explanation for this problem.
We know that the prime factorisation is a method that that we divide the given number with prime numbers starting from 2. If the number is divisible then we apply the same factorisation method for the quotient. We carry on this process until we get the quotient as a prime number.
But we know that the number 512 is obtained by raising the power of 2 to 9 that is
\[\Rightarrow 512={{2}^{9}}\]
Here, we can see that the number 512 is represented as the product of prime number 2 which will be the required representation.
Therefore the number 512 can be represented as the product of prime numbers as
\[\therefore 512={{2}^{9}}\]
Students should be able to understand the question and not write numbers like 4, 128, 64… since they are not prime factors.