Let $n$ be a positive integer with $n \leq 20$. The sum
$$1 + 2 + 3 + \cdots + n$$
is divisible by 9. Accordingly, what is the sum of the possible values of n?
A) 50
B) 52
C) 56
D) 60
E) 64

$\mathbf { a } < \mathbf { b } < \mathbf { c }$ are positive integers and
$$\begin{aligned} & \gcd ( a , b ) = 5 \\ & \gcd ( b , c ) = 4 \end{aligned}$$
Given this, what is the minimum value that the sum $\mathbf { a + b + c }$ can take?
A) 27
B) 35
C) 39
D) 45
E) 49

Let $\mathrm { a } , \mathrm { b }$ and c be prime numbers such that
$$\mathrm { ab } + \mathrm { ac } = 4 \mathrm { a } ^ { 2 } + 8$$
Given this, what is the product $\mathbf { a } \cdot \mathbf { b } \cdot \mathbf { c }$?

Let n be an integer greater than 2, and let the largest prime divisor of n be denoted by $\tilde{n}$. The terms of the sequence $(a_n)$ are defined for $n \geq 2$ as
$$a _ { n } = \left\{ \begin{aligned} 1 & , \tilde{n} < 10 \\ - 1 & , \tilde{n} > 10 \end{aligned} \right.$$
Accordingly, what is the sum $\sum _ { n = 15 } ^ { 30 } a _ { n }$?
A) 2
B) 3
C) 4
D) 5
E) 6

The sum of the prime divisors of a natural number A is;

• 3 less than the sum of the prime divisors of $12 \cdot A$.
• 5 less than the sum of the prime divisors of $70 \cdot A$.

Accordingly, what is the sum of the digits of the smallest value that A can take?
A) 4
B) 5
C) 6
D) 7
E) 8

$$1 ^ { 5 } + 2 ^ { 5 } + 3 ^ { 5 } + 4 ^ { 5 } + 5 ^ { 5 }$$
What is the remainder when this expression is divided by 7?
A) 4
B) 3
C) 2
D) 1
E) 0

Let $a$ and $b$ be distinct positive integers such that LCM(a,b) equals a prime number.
Accordingly,\ I. $a$ and $b$ are coprime numbers.\ II. The sum $a + b$ is an odd number.\ III. The product $\mathrm{a} \cdot \mathrm{b}$ is an odd number.
Which of the following statements are always true?\ A) Only I\ B) Only II\ C) Only III\ D) I and II\ E) II and III

Let p, r, and t be different prime numbers;

• Integer multiples of p form set A,
• Integer multiples of r form set B,
• Integer multiples of t form set C.

It is known that two of the numbers 220, 245, 330, and 350 are elements of the blue-colored set, and the other two are elements of the yellow-colored set. Accordingly, what is the sum $\mathbf { p } + \mathbf { r } + \mathbf { t }$?
A) 10
B) 14
C) 15
D) 21
E) 23

The three-digit natural number ABA divided by the two-digit natural number A1 gives a quotient of 13 and a remainder of 19.
Accordingly, what is the sum $A + B$?
A) 8
B) 9
C) 10
D) 11
E) 12

Let $p$ and $r$ be distinct prime numbers. The number $180 \cdot r$ is an integer multiple of the number $p$.
Accordingly, the prime number $p$ definitely divides which of the following numbers?
A) $12 \cdot r$
B) $18 \cdot r$
C) $20 \cdot r$
D) $30 \cdot r$
E) $45 \cdot r$

Let $m$ and $n$ be positive integers such that
$$\begin{aligned} & \gcd ( m , n ) + \text{lcm} ( m , n ) = 289 \\ & m + n \neq 289 \end{aligned}$$
What is the sum $m + n$?
A) 41
B) 43
C) 45
D) 47
E) 49