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[30 points] provide an explanation/show your work! Positive integer a has two different prime factors p and q (p<q) such that a = p*q. positive integer b is greater than a and the quotient a^2/b is an integer. How many possible values of b are there?

Circle the correct answer:
a) 2
b) 3
c) 4
d) 5
e) more than 5.

Thank you! I will give Brainliest to the best answer. :)​

30 points provide an explanationshow your work Positive integer a has two different prime factors p and q pltq such that a pq positive integer b is greater than class=

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konrad509

[tex]\dfrac{a^2}{b}[/tex] is an integer, where [tex]a,b\in\mathbb{Z_+}[/tex], therefore [tex]b|a^2[/tex] and [tex]b\leq a^2[/tex]

[tex]a=pq[/tex] therefore [tex]a^2=p^2q^2[/tex]

[tex]b>a\wedge b\leq a^2[/tex], therefore [tex]pq<b\leq p^2q^2[/tex]

So,

[tex]b\in\{p^2q,pq^2,p^2q^2\} \\|\{p^2q,pq^2,p^2q^2\}|=3 \implies \text{B}[/tex]

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