AMC8 2016

AMC8 2016 · Q24

AMC8 2016 · Q24. It mainly tests Remainders & modular arithmetic, Digit properties (sum of digits, divisibility tests).

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?

数字1、2、3、4和5各使用一次，写成五位数PQRST。三位数PQR能被4整除，三位数QRS能被5整除，三位数RST能被3整除。P是多少？

(A) 1 1

(B) 2 2

(D) 4 4

(E) 5 5

Answer

Correct choice: (A)

正确答案：（A）

Solution

Since QRS is divisible by 5, we know that S = 5. Since PQR is divisible by 4, we know that QR is 12, 32, or 24. So RST will be either 25T or 45T and divisible by 3. Using the available digits, 453 is the only number that is divisible by 3. So T = 3, R = 4, and P = 1.

由于 QRS 能被 5 整除，所以 S = 5。由于 PQR 能被 4 整除，所以 QR 为 12、32 或 24。因此 RST 将是 25T 或 45T，且需能被 3 整除。使用可用的数字，453 是唯一能被 3 整除的数。因此 T = 3，R = 4，P = 1。

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