If a natural number p has 8 factors, then which of the following cannot be the difference between the number of factors of p^3 and p?
Choose the correct choice from the following answers
A) 14
B) 30
C) 32
D) 56
E) None of these
p has 8 factors. That gives following possibilities:   a) p has one prime factor with power 7   =&gt; p^3 will have 22 factors. Difference with number of factors of p = 22-8= 
<gadget id="calculator">22-8</gadget>
<output>14</output>
14: A is ok   b) p has two prime factors with powers 3 and 1=&gt; p^3 will have 40 factors. Difference with number of factors of p =40-8= 
<gadget id="calculator">40-8</gadget>
<output>32</output>
32 : C is ok   c) p has three prime factors with powers 1 each=&gt; p^3 will have 64 factors. Difference with number of factors of p=64-8= 
<gadget id="calculator">64-8</gadget>
<output>56</output>
56 : D is ok   There are no other possibilities. Hence remaining answer choice B is not possible.   Answer 30
<result>B</result>
If a natural number p has 8 factors, then which of the following cannot be the difference between the number of factors of p^3 and p?