TITLE: Dividing up a fortune
QUESTION [5 upvotes]: I am trying to solve this below problem:

At his death, a millionaire left his 10 children a million dollars in cash, all in $\$100$, $\$10$, $\$1$ bills,10-cent, and 1-cent coins.  Show that there is a way for them to split the fortune into ten stacks of equal value.  (Note that this would not be true if there were $\$3$ bills).

So the best I can do is set this up as solving an equation in five variables. I want to find $a,b,c,d,e$ so that
$$ 
10^6 = a \cdot 100 + b \cdot 10 + c \cdot 1 + d \cdot 0.1 + e \cdot .01.
$$
I can try to take congruences mod $10$. The LHS is clearly $0$, and any multiple of $10$ is certainly $0$, but $\frac{1}{10}$ is not even defined in $\mathbb{Z}/10\mathbb{Z}$, since that would amount to division by zero, so this strategy does not work.
Can anyone give me a hint on how to get started?

REPLY [2 votes]: First, prove that the number of 1-cent coins is divisible by 10. This follows from the fact that "cents place" of \$1 million is zero. Then, WLOG, exchange the 1-cent coins for an equivalent number of 10-cent coins.
Then, prove that the number of 10-cent coins is divisible by 10. This follows from the fact that there are no 1-cent coins, and the "10-cent place" of \$1 million is zero. Then, exchange the 10-cent coins for an equivalent number of \$1 coins.
...
Then, prove there must be a multiple of 10 \$100 bills. This is because all the currency now is in \$100 bills.