Say I have a set $Q$ of integers between $0$ and $100$. As a constraint, the arithmetic sum of all elements adds up to $100$. We can interpret this as an "integer probability space".
Now, I wish to renormalize the elements of $Q$ to an integer between $0$ and $8$ (the use of a rounding function will be required). Can I ensure that the sum of the rescaled integers sum to $8$?
For instance; say the elements of $Q$ are $\{50,30,20\}$. First, let's verify that they sum to $100$: $50+30+20=100$, ok!. Now, I renormalize them to a scale on $0$ to $8$, and I get
$$ 50/100 \to 4/8\\ 30/100\to 2.4/8\\ 20/100\to 1.6/8 $$
To promote (demote?) the new elements to an integer, I $Round$ the values and I get $4/8,2/8,2/8$ which sums to $8$. Good!
However, If I instead chose a different renormalization scale (say $5$ instead of $8$), then the possibility exists that my rescale breaks the normalization condition. For instance, the elements would rescale as follows: $$ 50/100\to 2.5/5 \\ 30/100 \to 1.5/5 \\ 20/100 \to 1/5 $$ Rounding them I get $3/5+2/5+1/5=6/5$, and the normalization is broken.
In the case of $8$ it does not seem possible that the normalization condition be broken by rounding the re-scaling. We can see this because $0.5/8=6.25%$, $1.5/8= 18.75%$, etc. Since the mid value on the $8$-scale cannot produce integer values on the $100$-scale, it thus appears that the problem of breaking the normalization cannot occur as a result of the application of the rounding function.
Can someone see anything wrong with my analysis?