##### Number Theory: Place-Out Exam

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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 $$17?$$