Answer:
If statement(1) holds true, it is correct that [tex]\small \frac{r}{s}[/tex] is an integer.
If statement(2) holds true, it is not necessarily correct that [tex]\small \frac{r}{s}[/tex] is an integer.
Step-by-step explanation:
Given two positive integers [tex]r[/tex] and [tex]s[/tex].
To check whether [tex]\small \frac{r}{s}[/tex] is an integer:
Condition (1):
Every factor of [tex]s[/tex] is also a factor of [tex]r[/tex].
[tex]r \geq s[/tex]
Let us consider an example:
[tex]s = 5^2 \cdot 2\\r = 5^3 \cdot 2^2[/tex]
[tex]\dfrac{r}{s} = \dfrac{5^3\cdot2^2}{5^2\cdot2} = 10[/tex]
which is an integer.
Actually, in this situation [tex]s[/tex] is a factor of [tex]r[/tex].
Condition 2:
Every prime factor of s is also a prime factor of r.
(But the powers of prime factors need not be equal as we are not given the conditions related to powers of prime factors.)
Let
[tex]r = 2^2\cdot 5\\s =2^4\cdot 5[/tex]
[tex]\dfrac{r}{s} = \dfrac{2^3\cdot5}{2^4\cdot5} = \dfrac{1}{2}[/tex]
which is not an integer.
So, the answer is:
If statement(1) holds true, it is correct that [tex]\small \frac{r}{s}[/tex] is an integer.
If statement(2) holds true, it is not necessarily correct that [tex]\small \frac{r}{s}[/tex] is an integer.
