2017 AMC 10B 第 23 题

Question

令

N=123456789101112\dots4344

为把整数

1

到

44

依次写在一起形成的

79

位数。

N

除以

45

的余数是多少？

Let

N=123456789101112\dots4344

be the

79

-digit number that is formed by writing the integers from

1

to

44

in order, one after the other. What is the remainder when

N

is divided by

45?

1

4

9

18

44

答案：C

知识点：模运算中国剩余定理数字

难度评级：1660

小提示：

分别求模

5

和模

9

的余数。

Find the remainder modulo

5

and modulo

9

大提示：

再把这两个条件合并成模

45

的余数。

Combine them to get the remainder modulo

45

解答：

要找除以

45

的余数，必须分别找除以

5

和

9

的余数。除以

5

的余数就是个位数字除以

5

的余数，所以是

4

。

要找除以

9

的余数，通常找数字和。不过每个两位数除以

9

的余数与它的数字和相同，所以可以直接把

1

到

44

的整数相加，因为每个整数与它自己的数字和同余。这个和为

\dfrac{44\cdot45}{2}=990

，它是

9

的倍数。因此

N

是

9

的倍数。

这个数是

9

的倍数，并且除以

5

的余数为

4

，所以除以

45

的余数为

9

。

所以正确答案是 C。

To find the remainder when divided by

45,

we must find the remainder when divided by

5

and

9.

The remainder when divided by

5

is the remainder when the units digit is divided by

5,

making it

4.

To find the remainder when divided by

9,

we usually find the sum of the digits. However, each double digit number has the same remainder when divided by

9

as its digit sum, so we can just sum the integers from

1

to

44,

because each integer is congruent to its own digit sum. This sum is

\dfrac{44\cdot45}{2}=990,

which is a multiple of

9.

Thus,

N

is a multiple of

9.

Since it is a multiple of

9

and has a remainder of

4

when divided by

5,

the remainder when divided by

45

is

9.

Thus, the correct answer is C .

2017 AMC 10B 第 23 题

答案：C

难度评级：1660

小提示：

大提示：

解答：

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