2016 AMC 8 Problem 24 Number Theory

 Understand the Problem

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 $P$? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$
Source of the problem

2016 AMC 8 Problem 24

Topic

Number Theory

Difficulty Level
Easy
Suggested Book
Mathematical Circles

Start with hints

Do you really need a hint? Try it first!

We see that since QRS  is divisible by $5$$S$ must equal either $0$ or $5$, but it cannot equal $0$, so $S=5$ Similarly proceed .
 We notice that since  PQR   must be even, R must be either $2$ or $4$. However, when \( R = 2 \) we see that \( T  \equiv  2  (mod \ 3) \), which cannot happen because $2$ and $5$ are already used up; so $R=4$
This gives   $T \equiv 3 \pmod{4}$    meaning $T=3$Almost we are done . Try to think the rest .
 Now, we see that Q  could be either $1$ or $2$, but $14$ is not divisible by $4$, but $24$ is. This means that $R=4$ and  the P is clearly equal to 1.

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