Question

Express the following number as the product of the power of their prime factors
$2000$.

Answer 1

Hint: The first step is to find the factors of $2000$ and then find out the prime factors of $2000$. Then, we can write it in terms of exponents of the prime factors.

Complete step by step answer:
Before proceeding with the question, we must know how to find the factors of $2000$. We can find the factors by dividing $2000$ with the numbers by which it is divisible. If it is divisible by that number, then that number is the factor of $2000$. Prime factors are the factors which are prime numbers. Prime factorisation is finding which prime numbers multiply together to make the original number.

In this question, we have to express $2000$ as the product of prime factors.
Prime factors for $2000$ are found by dividing it with its least factor. We know that a number is divisible by 2 if the last digit of that number is 0, 2, 4, 6 or 8. Here, we have 2000 as the number and the last digit is 0, so it is divisible by 2. We have the first prime factor as 2. Now, for a number to be divisible by 3, the sum of its digits must be divisible by 3. The sum of digits of 2000 is 2, which is not divisible by 3. So, we move on to the next prime factor, 5. If the last digit of the number is 0 or 5, then it is divisible by 5. So, here 0 is the last digit of 2000, so we get 5 as the next prime factor.

Using the information above, we can write 2000 as,

$2000=2\times 2\times 2\times 2\times 5\times 5\times 5$

Clubbing the powers of the prime factors, we get

$\Rightarrow 2000={{2}^{4}}\times {{5}^{3}}$

Hence, $2000$ can be expressed as the product of the power of its prime factors, i.e. 2 and 5 as ${{2}^{4}}\times {{5}^{3}}$.


Note: Be careful while reading the question, we have been asked to find prime factors, not the factors. So, we have to find the least number which is the factor of the given number and that least number must also be a prime number. After getting the prime factors, we must not forget to write the prime factors in terms of exponential form.