AMC10 2013 A

AMC10 2013 A · Q19

AMC10 2013 A · Q19. It mainly tests Divisibility & factors, Primes & prime factorization.

In base 10, the number 2013 ends in the digit 3. In base 9, on the other hand, the same number is written as $(2676)_9$ and ends in the digit 6. For how many positive integers $b$ does the base-$b$ representation of 2013 end in the digit 3 ?

在10进制下，数字2013 以数字3结尾。而在9进制下，同一个数字写成$(2676)_9$，以数字6结尾。有多少个正整数$b$使得2013在$b$进制表示以数字3结尾？

(A) 6 6

(B) 9 9

(D) 16 16

(E) 18 18

Answer

Correct choice: (C)

正确答案：（C）

Solution

For the base-$b$ representation of 2013 to end in 3, $b>3$ and $b$ divides 2010=2013-3=2*3*5*67, 16 factors >3, minus 1? 16 total factors, minus those <=3:1,2,3 so 13.

要使2013在$b$进制表示以3结尾，需$b>3$且$b$整除$2010=2013-3=2\times3\times5\times67$，有16个因数大于3，减去那些$\leq3$的：1,2,3，故13个。

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