Question

In how many ways, the number 10800 can be resolved as a product of two factors?

Answer 1

- Hint-The main idea to solve this type of question is that we have to find all the factors of that given number because every factor of that number is bound to another factor so that if we multiply two factors, we will get back the same number. After finding all the factors, we just have to half the number of factors to get the answer.

Complete step-by-step solution -

Firstly, we will have to find the factors of the given number 10800
For finding the factors of a huge number will take us a lot of time. So we need to prime factorize the given number.
Let the prime factorization is
\[{a^{k1}} * {b^{k2}} * {c^{k3}} * ...... * {n^{kn}}\]
Where a, b, c ….n are prime numbers and
k1, k2, k3 ….kn are the powers to the prime numbers respectively. Then
The general formula for calculating the number of ways in which any number can be resolved as a product of two factors is
$\dfrac{{((k1 + 1) * (k2 + 1) * (k3 + 1) * ... * (kn + 1))}}{2}$ $if{\text{ }}((k1 + 1) * (k2 + 1) * (k3 + 1)...(kn + 1)){\text{ is even}}$
$\dfrac{{((k1 + 1) * (k2 + 1) * (k3 + 1) * ... * (kn + 1)) + 1}}{2}$ $if{\text{ }}((k1 + 1) * (k2 + 1) * (k3 + 1)...(kn + 1)){\text{ is odd}}$
After prime factorization of 10800, we got
${2^4} * {3^3} * {5^2}$
After applying the formula, the number of ways in which 10800 can be resolved as a product of two factors is
$ = ((4 + 1)*(3 + 1)*(2 + 1))/2$
$ = 60/2$
$ = 30$
$\therefore {\text{ In 30 ways, 10800 can be resolved as a product of two factors}}$
Note-The main point to notice in these types of questions is that when a number is a perfect square. For example, 36. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36 after applying formula, the result we get is $9/2$ but 4.5 does not make any sense. Let us manually make the pair of factors which are bounded with each other
$
  1*36 \\
  2*18 \\
  3*12 \\
  4*9 \\
  6*6 \\
$
We are getting 5 ways manually. This is happening due to the occurrence of perfect squares. So we have to add 1 in 9 to get the correct answer.