AMC10 2020 B

AMC10 2020 B · Q7

AMC10 2020 B · Q7. It mainly tests Primes & prime factorization, Parity (odd/even).

How many positive even multiples of 3 less than 2020 are perfect squares?

小于2020的有多少个正偶数3的倍数的完全平方数？

(A) 7 7

(B) 8 8

(D) 10 10

(E) 12 12

Answer

Correct choice: (A)

正确答案：（A）

Solution

In order for a perfect square $n^2$ to be even, $n$ must be a multiple of 2. Similarly, in order for $n^2$ to be a multiple of 3, $n$ must be a multiple of 3. Therefore $n$ must itself be a multiple of 6. Because $(7 \cdot 6)^2 = 1764 < 2020$ and $(8 \cdot 6)^2 = 2304 > 2020$, there are 7 such squares, namely $6^2, 12^2, 18^2, 24^2, 30^2, 36^2$, and $42^2$.

为了使完全平方$n^2$为偶数，$n$必须是2的倍数。同样，为了使$n^2$是3的倍数，$n$必须是3的倍数。因此$n$本身必须是6的倍数。因为$(7 \cdot 6)^2 = 1764 < 2020$且$(8 \cdot 6)^2 = 2304 > 2020$，因此有7个这样的平方数，即$6^2, 12^2, 18^2, 24^2, 30^2, 36^2$和$42^2$。

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