If $ \ p^3q^2=r^2s^3 \ $ , then according to the fundamental theorem of arithmatic

Question

Assume that $p,q,r,s$ are prime. If $$p^3q^2=r^2s^3,$$ then according to the fundamental theorem of arithmetic:

(i) $ \ q=s \ $

(ii) $ \ q >r \ $

(iii) $ \ p^3=s^2 \ $

(iv) $ \ p^3<s^2 \ $

Answer:

We have

$ p^3q^2=r^2s^3 \ \Rightarrow pppqq=rrsss \ ...........(1) $

If $ \ p=r=s \ $ , then from $ \ (1) $ , we get

$ rrsqq=rrsss \\ \Rightarrow qq=ss, \ \ (by \ \ left \ \ cancel) \\ \Rightarrow q=s $ ,

Thus option (i) is true.

what about the other options ?

Answer 1

According to the fundamental theorem of arithmetic we have that

$$ p^3q^2=r^2s^3 \implies p=s \,\land \, q=r$$

and then the implication is false for all the options but the options (i) is compatible with that for the particular case $p=s=q=r$ even if not true in general for the given condition.

Answer 2

$N= p^3q^2 =r^2s^3$.

If $p$ and $q$ are different primes than $N$ has precisely two prime factors one to the precise power of three and the other to the precise power of two and so $p = s$ and $q = r$.

If $p$ and $q$ are the same prime then $N$ has only one prime factor and it is to the fifth power so $p = q = r = s$.

So...

i) $q=s$. That could happen but is doesn't have to. If it does though then $p=q=r=s$. If $p \ne q$ then $s = p\ne q$.

ii) $q > r$. That's impossible as $q = r$ under all cases. However the FTA says nothing about the relative sizes of different primes. We can't say anything about whether $q$ is greater less or equal to $s=p$.

iii) $p^3 = s^2$ that'd be impossible for any two primes under any circumstances. Each number has a unique prime factorization and so if it has only one prime factor it must be to a specific power. It can't be one prime to the third and another prime squared.

iv) $p^3 < s^2$. Under all cases $p = s$ and as $p = s > 1$ then $p^3 > p^2 = s^2$ so this is impossible. Again if the primes were not equal we would no nothing about relative sizes.

So.....

i) is compatible but certainly not true, in the sense that it must be or is even likely to be.

ii)-iv) are all impossible.

So answer: none of them.