Professor Halfbrain has recently made a fascinating discovery on the two integers $x=101$ and $y=110$. The professor first observed that the decimal representation of $x$ is a permutation of the decimal representation of $y$. While this observation is quite straightforward (and some might even say: quite boring), the really fascinating part of Halfbrain's discovery was
- that the decimal digit sums of the numbers $2x$ and $2y$ coincide!
- and that the same holds true for the digit sums of $3x$ and $3y$!!
- and also for the digit sums of $4x$ and $4y$!!!
- and for the digit sums of $5x$ and $5y$!!!!
- and for the digit sums of $6x$ and $6y$!!!!!
- and for the digit sums of $7x$ and $7y$!!!!!!
- and for the digit sums of $8x$ and $8y$!!!!!!!
- and for the digit sums of $9x$ and $9y$!!!!!!!!
I am aware of the fact that this sounds almost unbelievable, but please trust me: it really is true, just check it yourself. Based on these experimental results, Professor Halfbrain has now formulated a theorem with eight statements. At the current moment, he is busily working on the proofs.
Halfbrain's big theorem: For any two positive integers $x$ and $y$ for which the decimal representation of $x$ is a permutation of the decimal representation of $y$, the following eight statements hold true:
(a). The digit sum of $2x$ equals the digit sum of $2y$.
(b). The digit sum of $3x$ equals the digit sum of $3y$.
(c). The digit sum of $4x$ equals the digit sum of $4y$.
(d). The digit sum of $5x$ equals the digit sum of $5y$.
(e). The digit sum of $6x$ equals the digit sum of $6y$.
(f). The digit sum of $7x$ equals the digit sum of $7y$.
(g). The digit sum of $8x$ equals the digit sum of $8y$.
(h). The digit sum of $9x$ equals the digit sum of $9y$.
Perhaps the professor was a little bit too excited about his discovery, and perhaps not each and every one of his eight statements will stand the test of time.
Can you help the professor in finding out which statements are true and which ones are false?