Numbers - Modular Arithmetic

SOURCE:COMPETITION
Number of Problems: 12. : (Book)

Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$ . For example, if $N = 749$ , Bernardo writes the numbers $10,444$ and $3,245$ , and LeRoy obtains the sum $S = 13,689$ . For how many choices of $n$ are the two rightmost digits of $S$ , in order, the same as those of $2N$ ?

$extbf{(A)} 5 qquad extbf{(B)} 10 qquad extbf{(C)} 15 qquad extbf{(D)} 20 qquad extbf{(E)} 25$

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 $3$ ?

$extbf{(A)} 6qquad extbf{(B)} 9qquad extbf{(C)} 13qquad extbf{(D)} 16qquad extbf{(E)} 18$

Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were 71,76,80,82, and 91. What was the last score Mrs. Walters entered?

$ext{(A)} 71 qquad ext{(B)} 76 qquad ext{(C)} 80 qquad ext{(D)} 82 qquad ext{(E)} 91$

How many distinct four-digit numbers are divisible by $3$ and have $23$ as their last two digits?

$extbf{(A) }27qquad extbf{(B) }30qquad extbf{(C) }33qquad extbf{(D) }81qquad extbf{(E) }90$

The first term of a sequence is $2005$ . Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the $2005^ ext{th}$ term of the sequence?

$mathrm{(A)} {{{29}}} qquad mathrm{(B)} {{{55}}} qquad mathrm{(C)} {{{85}}} qquad mathrm{(D)} {{{133}}} qquad mat...$

What is the units digit of $13^{2003}$ ?

$mathrm{(A) } 1qquad mathrm{(B) } 3qquad mathrm{(C) } 7qquad mathrm{(D) } 8qquad mathrm{(E) } 9$

Let $n$ be a $5$ -digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$ . For how many values of $n$ is $q+r$ divisible by $11$ ?

$mathrm{(A) } 8180qquad mathrm{(B) } 8181qquad mathrm{(C) } 8182qquad mathrm{(D) } 9000qquad mathrm{(E) } 9...$

What is the remainder when $3^0 + 3^1 + 3^2 + cdots + 3^{2009}$ is divided by 8?

$mathrm{(A)} 0qquadmathrm{(B)} 1qquadmathrm{(C)} 2qquadmathrm{(D)} 4qquadmathrm{(E)} 6$

Let $k={2008}^{2}+{2}^{2008}$ . What is the units digit of $k^2+2^k$ ?

$mathrm{(A)} 0qquadmathrm{(B)} 2qquadmathrm{(C)} 4qquadmathrm{(D)} 6qquadmathrm{(E)} 8$

10.

For $k > 0$ , let $I_k = 10ldots 064$ , where there are $k$ zeros between the $1$ and the $6$ . Let $N(k)$ be the number of factors of $2$ in the prime factorization of $I_k$ . What is the maximum value of $N(k)$ ?

$extbf{(A)} 6qquad extbf{(B)} 7qquad extbf{(C)} 8qquad extbf{(D)} 9qquad extbf{(E)} 10$

11.

The number obtained from the last two nonzero digits of $90!$ is equal to $n$ . What is $n$ ?

$extbf{(A)} 12 qquad extbf{(B)} 32 qquad extbf{(C)} 48 qquad extbf{(D)} 52 qquad extbf{(E)} 68$

12.

A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not 100 (such as 2012). The 200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7, 2012, a Tuesday. On what day of the week was Dickens born?

$extbf{(A)} ext{Friday}qquad extbf{(B)} ext{Saturday}qquad extbf{(C)} ext{Sunday}qquad extbf{(D)} ext{Mond...$