What is the count of positive integers that are multiples of 6 and also divisors of 72?

To determine how many positive integers are multiples of 6 and also divisors of 72, follow these steps:

1. 
   
   

   List the Divisors of 72:

   
   
   

   First, find all the positive divisors of 72. To do this, perform the prime factorization of 72:

   
   
   

   $72 = 2^3 \times 3^2$

   
   
   

   The number of divisors is given by multiplying one more than each of the exponents in the prime factorization:

   
   
   

   $(3 + 1) \times (2 + 1) = 4 \times 3 = 12 \text{ divisors}$

   
   
   

   Listing them out:

   
   
   

   $1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72$

   
   
   



2. 
   
   

   Identify Multiples of 6:

   
   
   

   From the list of divisors, identify which numbers are multiples of 6:

   
   
   

   $6, 12, 18, 24, 36, 72$

   
   
   

   These are the numbers that are divisible by both 2 and 3.

   
   
   



3. 
   
   

   Count the Valid Numbers:

   
   
   

   Count the numbers identified in the previous step:

   
   
   

   $6, 12, 18, 24, 36, 72 \quad \Rightarrow \quad 6 \text{ numbers}$

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