If three distinct integers are chosen at random, show that there will exist two among them, say $a$ and $b$, such that $30 | (a^3b-ab^3)$

Question

Problem:

If three distinct integers are chosen at random, show that there will exist two among them, say $a$ and $b$, such that $30 | (a^3b-ab^3)$

My work:

$a^3b-ab^3=ab(a+b) (a-b)$ and if $30 | (a^3b-ab^3)$, then each of $2,3,5$ divides $(a^3b-ab^3)$ since $2\cdot3\cdot5=30$ and $(2,3,5)=1$

Case for 2:

If 2 divides either $a$ or $b$, then all good. If not, then $a$ and $b$ are both odd and so $a+b$=sum of odds = even is divisible by 2. Done.

Case for 3:

If 3 divides a or b, good. Else, let:

1. $a=3k+1$ and $b=3k+1$. Then, $3|(a-b)$
2. $a=3k+1$ and $b=3k+2$. Then, $3|(a+b)$
3. $a=3k+2$ and $b=3k+2$. Then, $3|(a-b)$
4. $a=3k+2$ and $b=3k+1$. Then, $3|(a+b)$

So far, so good.

Case for 5:

Completely stuck here. Five possible remainders ($0,1,2,3,4$) seem to be impossible to do. Any hints?

Questions:

1. Need hints for last case.

2. Why do we need to pick three integers when we only need to check with two?

Answer 1

You can remove "distinct" to make it slightly stronger (but it's a trivial case). Notice $30=2\cdot 3\cdot 5$ and $2,3,5$ are primes. Your problem is: Among any three integers $a,b,c$, exist at least two $x,y\in\{a,b,c\}$ such that $2,3,5\mid xy(x+y)(x-y)$.

By Pigeonhole principle, either $abc\equiv 0\pmod{5}$ or $a\equiv \pm b\pmod{5}$ or $a\equiv \pm c\pmod{5}$ (because $a,b,c$ are three integers that belong to $0,1,2,-2,-1$ mod $5$), so exist $x,y\in\{a,b,c\}$ such that $5\mid xy(x+y)(x-y)$.

But then by Pigeonhole principle either $xy\equiv 0\pmod{3}$ or $x\equiv \pm y\pmod{3}$. Similarly, either $xy\equiv 0\pmod{2}$ or $x\equiv \pm y\pmod{2}$.

One generalization can be: Among any $k\ge 3$ integers $a_1,a_2,\ldots, a_k$ exist $x,y\in\{a_1,a_2,\ldots,a_k\}$ such that $6h\mid xy(x+y)(x-y)$, where $h\ge 5$ is any integer such that $\gcd(h,6)=1, \frac{h-1}{2}<k$.

Answer 2

Let $p,q$ be remainders when $5$ divides $a,b$ respectively. If any of the following holds:

1. $p=0$
2. $q=0$
3. $p=q$
4. $p+q = 5$

Then, $5|a$, $5|b$, $5|(a-b)$ or $5|(a+b)$ respectively.

So, now check for $(p,q)=$

1. $(1,2)$ Now, if $c = 5k+1$, then $5|(a-c)$, so repeat the question taking the two numbers $a,c$ noting that these two already answer the case for $5$. If, $c=5k+2$, then $5|(b-c)$, proceed similarly as above. If $c=5k+3$, then $5|(b+c)$, proceed similarly as above. If $c=5k+4$, then $5|(a+c)$, proceed similarly as above. If $c=5k$, then $5|c$ so take the numbers $a,c$ or $b,c$ and proceed similar to above.

2. For the remaining tuples, repeat similar to what we did in point 1 just above.

Q.E.D.