Numbers - Factor and Multiple

SOURCE:COMPETITION
Number of Problems: 21. : (Book)

Three positive integers are each greater than $1$ , have a product of $27000$ , and are pairwise relatively prime. What is their sum?

$extbf{(A)} 100qquad extbf{(B)} 137qquad extbf{(C)} 156qquad extbf{(D)}} 160qquad extbf{(E)} 165$

For how many positive integers $n$ is $n^2 - 3n + 2$ a prime number?

$mathrm{(A)} ext{none}qquadmathrm{(B)} ext{one}qquadmathrm{(C)} ext{two}qquadmathrm{(D)} ext{more than tw...$

Suppose that $n$ is the product of three consecutive integers and that $n$ is divisible by $7$ . Which of the following is not necessarily a divisor of $n$ ?

$extbf{(A)} 6 qquad extbf{(B)} 14 qquad extbf{(C)} 21 qquad extbf{(D)} 28 qquad extbf{(E)} 42$

Suppose that $n$ is the product of three consecutive integers and that $n$ is divisible by $7$ . Which of the following is not necessarily a divisor of $n$ ?

$extbf{(A)} 6 qquad extbf{(B)} 14 qquad extbf{(C)} 21 qquad extbf{(D)} 28 qquad extbf{(E)} 42$

Which of the following numbers is a perfect square?

$mathrm{(A) } 98! cdot 99! qquad mathrm{(B) } 98! cdot 100! qquad mathrm{(C) } 99! cdot 100! qquad mathrm{(D) ...$

10.

What is the probability that a randomly drawn positive factor of $60$ is less than $7$ ?

$mathrm{(A) } frac{1}{10}qquad mathrm{(B) } frac{1}{6}qquad mathrm{(C) } frac{1}{4}qquad mathrm{(D) } frac{...$

11.

The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?

$mathrm{(A) } 2qquad mathrm{(B) } 3qquad mathrm{(C) } 4qquad mathrm{(D) } 5qquad mathrm{(E) } 6$

12.

Let $n$ be the largest integer that is the product of exactly 3 distinct prime numbers $d$ , $e$ , and $10d+e$ , where $d$ and $e$ are single digits. What is the sum of the digits of $n$ ?

$mathrm{(A) } 12qquad mathrm{(B) } 15qquad mathrm{(C) } 18qquad mathrm{(D) } 21qquad mathrm{(E) } 24$

13.

What is the probability that an integer in the set ${1,2,3,...,100}$ is divisible by $2$ and not divisible by $3$ ?

$mathrm{(A) } frac{1}{6}qquad mathrm{(B) } frac{33}{100}qquad mathrm{(C) } frac{17}{50}qquad mathrm{(D) } ...$

14.

15.

16.

17.

How many positive cubes divide $3! cdot 5! cdot 7!$ ?

$mathrm{(A) } 2qquad mathrm{(B) } 3qquad mathrm{(C) } 4qquad mathrm{(D) } 5qquad mathrm{(E) } 6$

18.

Suppose that $m$ and $n$ are positive integers such that $75m = n^{3}$ . What is the minimum possible value of $m + n$ ?

$ext{(A)} 15 qquad ext{(B)} 30 qquad ext{(C)} 50 qquad ext{(D)} 60 qquad ext{(E)} 5700$

19.

20.

21.

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 $N(k)$ ?

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