Six bags contain 18, 19, 21, 23, 25, and 34 coins respectively, one of which is entirely filled with fake coins, while the other five bags do not contain any fake coins. Xiao Cong took three bags, and Xiao Min took the other two bags, leaving the bag with the fake coins. If the total number of coins Xiao Cong got is twice that of Xiao Min, then how many fake coins are there?
Number Theory, Congruence.
Since the total number of coins Xiao Cong received is twice that of Xiao Min;
That is, the total number of coins Xiao Cong received is 2 times that of Xiao Min;
Therefore, the total number of coins Xiao Cong and Xiao Min received is 3 times that of Xiao Min;
That is, the total number of coins Xiao Cong and Xiao Min received is a multiple of 3;
Thus, the total number of coins in the six bags and the number of coins in the remaining fake coin bag are congruent modulo 3;
$$
\begin{array}{l}
18 \equiv 0(\bmod 3), 19 \equiv 1(\bmod 3), 21 \equiv 0(\bmod 3), 23 \equiv 2(\bmod 3), 25 \equiv 1(\bmod 3), 34 \equiv 1(\bmod 3) ; \\
19+19+21+23+25+34 \equiv 0+1+0+2+1+1=5 \equiv 2(\bmod 3) ;
\end{array}
$$

Therefore, the fake coins are 23 in number.