AMC8 1991

AMC8 1991 · Q13

AMC8 1991 · Q13. It mainly tests Primes & prime factorization, Digit properties (sum of digits, divisibility tests).

How many zeros are at the end of the product $25 \times 25 \times 25 \times 25 \times 25 \times 25 \times 8 \times 8 \times 8$?

$25 \times 25 \times 25 \times 25 \times 25 \times 25 \times 8 \times 8 \times 8$这个乘积末尾有多少个零？

(A) 3 3

(B) 6 6

(D) 10 10

(E) 12 12

Answer

Correct choice: (C)

正确答案：（C）

Solution

Since $2 \times 5 = 10$, each zero at the end of the product comes from a product of 2 and 5 in the prime factorization. Since $25 = 5 \times 5$ and $8 = 2 \times 2 \times 2$, it follows that there are fourteen factors of 5 and nine factors of 2. This yields 9 pairs of $2 \times 5$ and results in 9 zeros at the end of the product.

因为$2 \times 5 = 10$，乘积末尾每个零来自质因数分解中的2和5的乘积。因为$25 = 5 \times 5$且$8 = 2 \times 2 \times 2$，有十四個5的因子和九個2的因子。这产生9对$2 \times 5$，结果乘积末尾有9个零。

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