I had a deck of cards but I lost a few. When I play with 3 other people, I have three cards left. When I play with 2 other people 2 cards are left. When I play with 4 other people, I have 2 cards left. How many cards have I lost?
+6251 \(\text{let the number of cards left in the deck be $N$}\\ N \pmod{4} = 3\\ N \pmod{3} = 2\\ N \pmod{5} = 2\\ N \approx 52,~N < 52\)
\(N =4k+3\\ 4k+3 \pmod{3} = 2\\ k +(3k+3) \pmod 3 = 2\\ k \pmod{3} = 2\\ k = 3j + 2\\ N = 12j + 11\\ 12j + 11 \pmod{5} = 2\\ 2j+1 + (10j + 10) \pmod{5} = 2\\ 2j+1 \pmod{5} = 2\\ 2j \pmod{5}=1\\ 2j \pmod{5}=6\\ j \pmod{5} = 3\\ j = 5i+3\\ N = 60i+36+11 = 60i + 47\\ N=47 \)
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