Question

If $M = {2^2} \times {3^5}$ and $N = {2^2} \times {3^4}$ then number of factors of N that are common to factors of M is:
A) 8
B) 5
C) 18
D) 15

Answer 1

Hint: In order to find the number of common factors we need to find the number of factors of each element first. We will use the standard formula for counting the number of factors.
Total number of factors of $T = \left( {a + 1} \right)\left( {b + 1} \right)$
$T$ be a natural number of the form:
$T = {X^a} \times {Y^b}$
Where $X$ and $Y$ are the prime factors and $a$ and $b$ are their powers

Complete step by step solution:
We are given the following elements:
 $M = {2^2} \times {3^5}$
$N = {2^2} \times {3^4}$
We need to find a number of common factors for both $M$ and $N$.
Therefore first we need to find the total number of factors of each element. To find the total number of factors we first need to write the given element as the multiples of prime factors and then apply the formula for the number of factors which can be described as:
Let $T$ be a natural number for which we need to find a number of factors. First, we need to write $T$ as the product of prime numbers by prime factorization method which can be represented as:
$T = {X^a} \times {Y^b}$
Where X and Y are the prime factors and a and b are their powers. Then,
Total number of factors of
Now applying this formula for $M$ we get the values of $a$ and $b$ as $2$ and $5$ respectively, therefore,
The total number of factors of $M$ is:
$
  M = \left( {2 + 1} \right) \times \left( {5 + 1} \right) \\
  {\text{ = 3}} \times 6 \\
  {\text{ = 18 }} \\
 $
Now applying the same formula for $N$ we get the values of $a$ and $b$ as $2$ and $4$ respectively, therefore,
The total number of factors of $N$ are:
$
  N = \left( {2 + 1} \right) \times \left( {4 + 1} \right) \\
  {\text{ = 3}} \times 5 \\
  {\text{ = 15}} \\
 $
Now, since both $M$ and $N$ have the same prime factors, therefore the number of common factors will be equal to the number of factors of the element which has less number of factors, and since $N$ has less number of common factors in comparison to $M$ therefore,
The number of common factors of $M$ and $N$ equal to the number of factors of $N$.

Hence, number of common factors of M and N$ = 15$. So, option (D) is correct.

Note:
In order to find the factors of an element we first need to factorize the element as the product of prime factors using the prime factorization method.