Question

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 $P?$

$1$

$2$

$3$

$4$

$5$

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Written solution:

Since $QRS$ is divisible by $5,$ we know that $S = 5.$

Since $PQR$ is divisible by $4,$ $QR$ equals either $12, 24,$ or $32.$

If $QR=12,$ then $R=2,$ so $RST=25T$ cannot be divisible by $3$ using the remaining digits. If $QR=32,$ then $R=2$ again, giving the same obstacle. Thus $QR=24,$ and then $RST=45T.$ Among the remaining digits, only $T=3$ makes $453$ divisible by $3$ .

Therefore, $PQRST = 12453.$

Thus, A is the correct answer.

2016 AMC 8 Problem 24

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Written solution:

Problem 24 in Other Years