AMC12 2015 A

AMC12 2015 A · Q15

AMC12 2015 A · Q15. It mainly tests Decimals, Primes & prime factorization.

What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{123456789}{2^{26}\cdot 5^{4}}$ as a decimal?

将分数 $\frac{123456789}{2^{26}\cdot 5^{4}}$ 表示为小数时，小数点右边最少需要多少位数字？

(A) 4 4

(B) 22 22

(D) 30 30

(E) 104 104

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): The numerator and denominator of this fraction have no common factors. To express the fraction as a decimal requires rewriting it with a power of 10 as the denominator. The smallest denominator that permits this is $10^{26}$: $$ \frac{123\,456\,789}{2^{26}\cdot 5^{4}} = \frac{123\,456\,789\cdot 5^{22}}{2^{26}\cdot 5^{4}\cdot 5^{22}} = \frac{123\,456\,789\cdot 5^{22}}{10^{26}}, $$ so the numeral will have 26 places after the decimal point. In fact $$ \frac{123\,456\,789}{2^{26}\cdot 5^{4}} = 0.00294343922138214111328125. $$

答案（C）：这个分数的分子和分母没有公因数。要把该分数表示成小数，需要把它改写成分母为 10 的幂。允许这样做的最小分母是 $10^{26}$： $$ \frac{123\,456\,789}{2^{26}\cdot 5^{4}} = \frac{123\,456\,789\cdot 5^{22}}{2^{26}\cdot 5^{4}\cdot 5^{22}} = \frac{123\,456\,789\cdot 5^{22}}{10^{26}}, $$ 因此该小数点后会有 26 位。事实上 $$ \frac{123\,456\,789}{2^{26}\cdot 5^{4}} = 0.00294343922138214111328125. $$

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