AMC12 2015 B

AMC12 2015 B · Q6

AMC12 2015 B · Q6. It mainly tests Basic counting (rules of product/sum), Parity (odd/even).

Back in 1930, Tillie had to memorize her multiplication facts from $0\times0$ through $12\times12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd?

1930年，Tillie需要记忆从$0\times0$到$12\times12$的乘法事实。她得到的乘法表有标记因数的行和列，积形成了表格的主体部分。主体表格中的数字有多少比例是奇数？（结果保留到百分位）

(A) 0.21 0.21

(B) 0.25 0.25

(D) 0.50 0.50

(E) 0.75 0.75

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): There are $13\cdot 13=169$ entries in the body of the table. An entry is odd if and only if both its row factor and its column factor are odd. There are $6$ odd whole numbers between $0$ and $12$, so there are $6\cdot 6=36$ odd entries in the body of the table. The required fraction is $\frac{36}{169}=0.213\ldots\approx 0.21$.

答案（A）：表格主体中共有 $13\cdot 13=169$ 个条目。一个条目为奇数当且仅当它的行因子和列因子都为奇数。$0$ 到 $12$ 之间有 $6$ 个奇的整数，因此表格主体中有 $6\cdot 6=36$ 个奇数条目。所求分数为 $\frac{36}{169}=0.213\ldots\approx 0.21$。

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