1. Problem: Given $m \cdot n = 24$ and $p_t$ is a prime number, find $p_t$ from options 38, 46, 73, 83. Since $p_t$ is prime and options are 38, 46, 73, 83, only 73 and 83 are prime. No direct relation to $m \cdot n=24$ is given, so $p_t$ must be prime from options: answer is 73. 2. Problem: Find the minimal number of equal cubes to divide a parallelepiped with sides 10 cm, 15 cm, 20 cm. Find the greatest common divisor (GCD) of the sides: $\gcd(10,15,20) = 5$ cm. Each cube has side 5 cm. Number of cubes = $\frac{10}{5} \times \frac{15}{5} \times \frac{20}{5} = 2 \times 3 \times 4 = 24$. 3. Problem: If integer $x$ divisible by 4, which of $x-4$, $x-3$, $x-1$, $x+3$ leaves remainder 1 when divided by 4? Since $x \equiv 0 \pmod{4}$: - $x-4 \equiv 0 - 0 = 0$ mod 4 - $x-3 \equiv 0 - 3 = 1$ mod 4 - $x-1 \equiv 3$ mod 4 - $x+3 \equiv 3$ mod 4 Answer: $x-3$. 4. Problem: Minimal number of squares to cut a 30 cm by 21 cm rectangle. Find GCD of sides: $\gcd(30,21) = 3$ cm. Number of squares = $\frac{30}{3} \times \frac{21}{3} = 10 \times 7 = 70$. 5. Problem: Ajara gave children 4 cookies each, then remaining cookies 5 each, no cookies left. Find possible total cookies. Let number of children be $k$. Total cookies = $4k + 5k = 9k$. Options: 45, 50, 60, 70. Only 45 and 60 divisible by 9. Answer: 45 or 60, but only 45 is option A, so 45. 6. Problem: Find remainder of $2020^n + 2019^{5n}$ mod 3. Calculate mod 3: $2020 \equiv 1$ mod 3, $2019 \equiv 0$ mod 3. So $2020^n \equiv 1^n = 1$ mod 3. $2019^{5n} \equiv 0^{5n} = 0$ mod 3. Sum mod 3 = $1 + 0 = 1$. Answer: 1. 7. Problem: If $m$ mod 15 = 7, find remainder of $m^2 + 3m$ mod 15. Calculate: $m^2 + 3m \equiv 7^2 + 3 \times 7 = 49 + 21 = 70$ mod 15. $70 \mod 15 = 10$. Answer: 10. 8. Problem: If integer $a$ mod 11 = 2, which of $a-9$, $a+2$, $a+8$, $a+9$ divisible by 11? Calculate each mod 11: $a-9 \equiv 2 - 9 = -7 \equiv 4$ mod 11 $a+2 \equiv 2 + 2 = 4$ mod 11 $a+8 \equiv 2 + 8 = 10$ mod 11 $a+9 \equiv 2 + 9 = 11 \equiv 0$ mod 11 Answer: $a+9$. 9. Problem: If $a$ mod 5 = 2 and $b$ mod 5 = 3, find remainder of $(a+b)$ mod 5. Sum mod 5 = $2 + 3 = 5 \equiv 0$. Answer: 0. 10. Problem: Sultan divides candies by 4 and 5 with no remainder. Find possible number of candies. Number must be divisible by LCM(4,5) = 20. Options: 45, 50, 60, 70. Only 60 divisible by 20. Answer: 60.