This is a riddle my friend gave me and we don't know the answer so we would like some help. The task is to use all of the numbers $0,1,2,…,9$ once each to get a sum of $100$ only using the plus sign.
You can do whatever you want with the numbers so long the plus sign and only that is included. For example, one could concatenate and use the number $12$ or $23$.
Multiplications or exponents or other signs are not allowed.
Best Answer
It is impossible, since the sum will always be divisible by $9$. Note that for a number to be divisible by $9$, the sum of its digits must be divisible by $9$; thus, we know that the solution must be divisble by $9$ as $1+\cdots+9=45$ is divisble by $9$, and so it can never be $100$.
To comment on the problem where we can use a decimal point - we have the same problem. Multiply both sides by a power of $10^k$ that makes the solution $x$ integers only - and $10^kx\equiv 1^kx\equiv x\equiv 100\equiv 1\mod 9$. But still the solution multiplied by $10^k$ only adds numbers with digits $1,\cdots,9$ and a number of $0$'s, so $10^kx\equiv 0\mod 9$, which is again a contradiction.
A good website to visit on related problems is cut-the-knot.