Which of the following is true?

Question

Let $\hspace{0.2cm}$$p,q,r$$\hspace{0.2cm}$ be prime numbers greater than 100,then which of the following is true?

$3|p^2+q^2+r^2$

$q|p^5$

There exists integers $x,y$ such that $\hspace{0.2cm}$$px+qy=r$

My work:The first result I know is true for all primes $p,q,r$ which are not divisible by $3$.

2nd is wrong,take $p=101,q=103$

About 3rd I do not have any idea.

Answer 1

If $p = q$ The second one is right.

Proof: Trivial, $q = p \vert p⁵$

If $p \neq q$ The third one is right.

Proof: $gcd(p,q)= 1$.With the extended Euclid Algorythm for every $a,b\in \mathbb{N}$ you find integers $m$ and $n$ that hold $m \cdot a+n \cdot b=gcd(a,b)$. Find those $m,n$ for $p=a, q = b$, then you have: $$(r \cdot m) \cdot p + (r \cdot n) \cdot q = r \cdot (m \cdot p + n \cdot q) = r \cdot gcd(p,q) = r \cdot 1 =r $$

Answer 2

1.) We have $p^2\equiv 1 (3)$ for $3\nmid p$ by Fermat, hence $p^2+q^2+r^2\equiv 1+1+1\equiv 0(3)$ for $3\nmid p,q,r$. Since $p,q,r>3$ we obtain $3\mid (p^2+q^2+r^2)$.

2.) $q\mid p^5$ iff $q\mid p$ for primes $p,q$, i.e., for $p=q$.

3.) If $p$ and $q$ are distinct, then the extended Euclidean algorithm yields integers $k,\ell$ with $kp+\ell q=1$. Now multiply with $r$. For $p=q$ we need to solve $xp+yp=r$, which implies $p\mid r$, and hence $p=q=r$. Then for example $3p-2p=p$ with $x=3$, $y=-2$.