##### 数论工具: 分级测试

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1.

How many choices for the digit A make the number 86A40 a multiple of 8?

2.

What choice of the digit A makes the number 135A9 have remainder 3 when divided by 9?

3.

What is the remainder when

$100 \times 101 \times 102 \times 103$

is divided by $11?$

4.

Find the last digit of:

$2222^{3141592}$

5.

Let $D(n)$ equal the number of factors of an integer $n.$ For how many values of $n$ from $1$ to $300$ is $D(n)$ odd?

6.

How many integers less than or equal to $255$ are relatively prime to $255?$

7.

Find the smallest positive integer $a$ that makes $\displaystyle \frac{a}{105}$ a terminating decimal.

8.

How many numbers from $1$ through $105$ have a remainder of $2$ when divided by $3$ and a remainder of $3$ when divided by $7?$

9.

The product of all the factors of $156$ has a prime factorization of $2^a b,$ where $b$ is an odd integer.

What is the value of $a?$

10.

What is the smallest four-digit number that has a remainder of $1$ when divided by $3,$ a remainder of $2$ when divided by $13,$ and a remainder of $2$ when divided by $17?$