Answer:
The 52th card that was dealt was an 8.
Step-by-step explanation:
The total value of the entire deck is [tex]4(1 + 2 + \cdots + 9 + 10 \cdot 3) = 4(\frac{10 \cdot 11}{2} + 20) = 220 + 80 \equiv 0 \mod 10.[/tex]
The total value of the 51 cards face-up is congruent to [tex]0 + 3 + 6 + 1 + 10 + 10 + 4 + 8 = 42 \equiv 2 \mod 10.[/tex]
The 52th card must therefore be an 8 for the total value of the entire deck to be 8 mod 10. (Or, rather, we have that the last card must be [tex]0-2 \equiv 8 \mod 10[/tex] and the only card that is congruent to that value modulo 10 is the 8.)
