Riddles, brain puzzles and mathematical problems

[BAndrew]

Prove that for any (natural) number of the form abcabc (eg 123123) is divisible with 1001 and also that the result of the division is abc.

[Froge]

NOTE 2: The solution is really, really short, but elegant.

q1 = kq2
q2 = kq3
...
qn-1 = kqn

substituting the above equations into each other

q1 = qn(k^(n-1))
q1/qn = k^(n-1)

[BAndrew]

@Froge

Correct!

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A more difficult version of this:

If

• f(3ax) / f(ax) = k
  

Then calculate

f(243x)/f(x)

It doesn't require any theorem or something like that. Just a basic knowledge of what a function is. The idea is the same with the previous problem (it depends on how you solved the previous riddle though).

[Froge]

more difficult? but (fun)ctions are more fun

[BAndrew]

more difficult? but (fun)ctions are more fun

It is both funny and difficult. Difficult is relevant of course. It may be easy for someone.

PS. Nice pun there!

[BAndrew]

You want to send a valuable object to a friend. You have a box which is more than large enough to contain the object. You have several locks with keys. The box has a locking ring which is more than large enough to have a lock attached. But your friend does not have the key to any lock that you have. How do you do it? Note that you cannot send a key in an unlocked box, since it might be copied.

NOTE: Your friend is intelligent and can understand your plan even if you can't communicate.

[Froge]

send the object

[BAndrew]

send the object

Good job! You just got your valuable item stolen.

[Froge]

So it has to be sent in a box?

[BAndrew]

So it has to be sent in a box?

Yes. You have to protect it with a lock, but then your friend doesn't have the key to unlock it. How do you do it?