The digits $1$, $2$, $3$, $4$, and $5$ are each used once to write a five-digit number $PQRST$. The three-digit number $PQR$ is divisible by $4$, the three-digit number $QRS$ is divisible by $5$, and the three-digit number $RST$ is divisible by $3$. What is $P$?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$
Solution 1 (Modular Arithmetic)
We see that since $QRS$ is divisible by $5$, $S$ must equal either $0$ or $5$, but it cannot equal $0$, so $S=5$. We notice that since $PQR$ must be even, $R$ must be either $2$ or $4$. However, when $R=2$, we see that $T \equiv 2 \pmod{3}$, which cannot happen because $2$ and $5$ are already used up; so $R=4$. This gives $T \equiv 3 \pmod{4}$, meaning $T=3$. Now, we see that $Q$ could be either $1$ or $2$, but $14$ is not divisible by $4$, but $24$ is. This means that $Q=2$ and $P=\boxed{\textbf{(A)}\ 1}$.
~CHECKMATE2021

Solution 2
We know that out of $PQRST,$ $QRS$ is divisible by $5$. Therefore $S$ is obviously 5 because $QRS$ is divisible by 5. So we now have $PQR5T$ as our number. Next, let's move on to the second piece of information that was given to us. $RST$ is divisible by 3. So, according to the divisibility by 3 rule, the sum of $RST$ has to be a multiple of 3. The only 2 big enough are 9 and 12 and since 5 is already given. The possible sums of $RT$ are 4 and 7. So, the possible values for $R$ are 1,3,4,3 and the possible values of $T$ are 3,1,3,4. So, using this we can move on to the fact that $PQR$ is divisible by 4. So, using that we know that $R$ has to be even so 4 is the only possible value for $R$. Using that we also know that 3 is the only possible value for 3. So, we have $PQRST$ = $PQ453$ so the possible values are 1 and 2 for $P$ and $Q$. Using the divisibility rule of 4 we know that $QR$ has to be divisible by 4. So, either 14 or 24 are the possibilities, and 24 is divisible by 4. So the only value left for $P$ is 1.  $P=\boxed{\textbf{(A)}\ 1}$.
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Solution 3 (Divisibility Rules)
We know that $QRS$ is divisible by $5$, so $S$ would be either $5$ or $0$. However, $0$ is not a choice, so $S=5$. Also, $PQR$ is divisible by $4$, so this means that $QR$ is $12$, $32$, $24$, or $52$. If $R=2$, then $T$ has to be $2$ or $5$ ($RST$ is divisible by $3$), but both are taken. So, $R=4 \Rightarrow QR=24$. $R+S+T$ must equal $9$ or $12$, but because $4+5=9$, $R+S+T=12 \Rightarrow T=3$. This leaves $P=\boxed{\textbf{(A) }1}$
~CHECKMATE2021

Solution 4 (Lucky and Fast)
We can simply try each of the answer choice, and we will see which one works. Trying $P=\boxed{\textbf{(A) }1}$, if $PQR$ is divisible by $4$, $QR$ must be divisible by four. Therefore, $QR$ can only be $24$, $52$, or $32$. However, since $QRS$ is divisible by $5$, $S = 5$, so $QR$ cannot be $52$. When $QR = 32$, $R = 2$, the last requirement cannot be satisfied because $R + S + T = 2 + 4 + 5 = 11$, and $11$ is not divisible by $3$. However, when $QR = 24$, $R = 4$, the last requirement can be satisfied. Hence, we can see that when $P=\boxed{\textbf{(A) }1}$, there is one way to satisfy all three requirements, leading to a conclusion that $P$ is $\boxed{\textbf{(A) }1}$.
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