Question

How many non-prime factors are in the number \[N = {2^5} \times {3^7} \times {9^2} \times {11^4} \times {13^3}.\]

Answer 1

Hint: In order to solve this problem first we have to calculate the total number of factors in the given expression we will calculate it by adding 1 to each exponent of a prime number and multiplying all of them, further to get the total number of non-prime factors we will subtract the number of prime factors from the total number of factors.

Complete step-by-step solution:
Prime numbers are basically the whole number greater than one that cannot be made by multiplying other whole numbers. Various examples of prime numbers are 2, 3, 5, 7, 11, etc. Here, we can see that the odd number 9 is not a prime number; this is because the odd number 9 can be written as the multiplication of 3 with 3.
Prime factors of number N are simply the prime numbers which when multiplied together gives the number N. Non-prime factors are the numbers that are not prime numbers (i.e., the numbers which have more than two factors including 1 and the number itself).
Given expression is \[N = {2^5} \times {3^7} \times {9^2} \times {11^4} \times {13^3}.\]
Here in the above equation, \[9\]can also be written as \[{3^2}\]
So expression becomes as
\[N = {2^5} \times {3^{11}} \times {11^4} \times {13^3}.\]
Now to calculate total number of factors we will add \[1\] to each exponent of prime number and multiplying all of them, so we have
Total number of factors \[ = {\text{ }}\left( {{\text{ }}5 + 1} \right){\text{ }} \times {\text{ }}\left( {11 + 1} \right){\text{ }} \times {\text{ }}\left( {4 + 1} \right){\text{ }} \times {\text{ }}\left( {{\text{ }}3 + 1} \right)\]
Further by simplifying above equation we have
\[6 \times 12 \times 5 \times 4 = 1440\]
Now here number of prime factors are \[4{\text{ }}\left( {{\text{ }}2,3,11,13} \right)\]
Therefore, number of non prime factors = Total number of factors \[ - \] total number of prime factors
By substituting the values we obtain \[1440 - 4 = 1436\]
Number of non prime factors = 1436
Hence, the total number of non prime factors is 1436.

Note: In these types of problems, we will write the given number in terms of the prime factors of any number. The number of non-prime factors is equal to the number of total factors minus the number of prime factors. A prime number is a naturally occurring number greater than 1 and is not a combination of two smaller numbers. A composite number is considered a real number greater than 1 but is not a prime.