2. $$ p $$ and $$ q $$ are two prime numbers and $$ p
eq q $$. (i) If 3 divides $$ p $$ evenly, then what is $$ p $$ ? (ii) If 5 divides $$ p $$ evenly, then what is $$ p $$ ? (iii) If 3 divides $$ p q $$ evenly, then what can you say about $$ p $$ and $$ q $$ ? (iv) If 3 divides $$ a^{2} $$ evenly and $$ a^{2}=(p q)^{2} $$, then explain why 3 divides $$ a $$ evenly.

Question

2. $$ p $$ and $$ q $$ are two prime numbers and $$ p 
eq q $$. (i) If 3 divides $$ p $$ evenly, then what is $$ p $$ ? (ii) If 5 divides $$ p $$ evenly, then what is $$ p $$ ? (iii) If 3 divides $$ p q $$ evenly, then what can you say about $$ p $$ and $$ q $$ ? (iv) If 3 divides $$ a^{2} $$ evenly and $$ a^{2}=(p q)^{2} $$, then explain why 3 divides $$ a $$ evenly.

UpStudy AI · Accepted Answer

Let's analyze each part of the question step by step.

(i) If 3 divides $$ p $$ evenly, then what is $$ p $$?

Since $$ p $$ is a prime number and 3 is a prime number, the only prime number that is divisible by 3 is 3 itself. Therefore, if 3 divides $$ p $$ evenly, then $$ p = 3 $$.

(ii) If 5 divides $$ p $$ evenly, then what is $$ p $$?

Similar to the previous case, since $$ p $$ is a prime number and 5 is also a prime number, the only prime number that is divisible by 5 is 5 itself. Therefore, if 5 divides $$ p $$ evenly, then $$ p = 5 $$.

(iii) If 3 divides $$ p q $$ evenly, then what can you say about $$ p $$ and $$ q $$?

If 3 divides $$ p q $$ evenly, then at least one of the prime numbers $$ p $$ or $$ q $$ must be divisible by 3. This is because if neither $$ p $$ nor $$ q $$ were divisible by 3, then their product $$ p q $$ would also not be divisible by 3. Therefore, either $$ p = 3 $$, $$ q = 3 $$, or both.

(iv) If 3 divides $$ a^{2} $$ evenly and $$ a^{2}=(p q)^{2} $$, then explain why 3 divides $$ a $$ evenly.

This follows from a property of prime numbers and divisibility. If a prime number $$ r $$ divides $$ n^2 $$ (where $$ n $$ is any integer), then $$ r $$ must also divide $$ n $$. In this case, since 3 divides $$ a^2 $$ evenly, it must also divide $$ a $$ evenly. Therefore, if $$ a^2 = (pq)^2 $$ and 3 divides $$ a^2 $$, then 3 must also divide $$ a $$.
(i) $$ p = 3 $$ if 3 divides $$ p $$ evenly. (ii) $$ p = 5 $$ if 5 divides $$ p $$ evenly. (iii) If 3 divides $$ p q $$ evenly, then either $$ p = 3 $$, $$ q = 3 $$, or both. (iv) If 3 divides $$ a^{2} $$ and $$ a^{2}=(p q)^{2} $$, then 3 divides $$ a $$ evenly.

and are two prime numbers and .
(i) If 3 divides evenly, then what is ?
(ii) If 5 divides evenly, then what is ?
(iii) If 3 divides evenly, then what can you
say about and ?
(iv) If 3 divides evenly and , then
explain why 3 divides evenly.

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