Thank you for uploading the image. I will now extract the mathematical content and solve or analyze the tasks presented. Give me a moment to process the text.Here’s the breakdown and solution of the tasks in the uploaded image:

Zadanie 8

Problem: Check the truthfulness of the statements.

1. Statement: A three-digit natural number $n$, when divided by 7, leaves a remainder of 6.

   • Analysis: If $n \mod 7 = 6$, then $n = 7k + 6$. For $n$ to be three-digit, $7k + 6 \in [100, 999]$.
     • Solve $7k + 6 \geq 100$: $7k \geq 94$ → $k \geq 14$.
     • Solve $7k + 6 \leq 999$: $7k \leq 993$ → $k \leq 141$.
     • Valid $k \in [14, 141]$, so the statement is true (P).

2. Statement: The number $2n + 1$ is divisible by 7.

   • Analysis: For $2n + 1$ to be divisible by 7, $2n + 1 \equiv 0 \pmod{7}$. Rearrange: $2n \equiv -1 \pmod{7}$, or $2n \equiv 6 \pmod{7}$, as $-1 \equiv 6$. Dividing by 2: $n \equiv 3 \pmod{7}$.
   • $n$ must satisfy $n = 7k + 3$. Substitute into $2n + 1 = 2(7k + 3) + 1 = 14k + 7$, which is divisible by 7. True (P).

Zadanie 9

Problem: Analyze properties of the number 2023.

1. Statement: 2023 can be expressed as the product of the first prime numbers $7 \cdot 17 \cdot 17$.

   • Analysis: $7 \cdot 17 \cdot 17 = 2023$, verified. True (P).

2. Statement: 2023 has exactly six natural divisors.

   • Analysis: The prime factorization $2023 = 7 \cdot 17^2$. The number of divisors is $(1 + 1)(2 + 1) = 6$. True (P).

3. Statement: 2023 has exactly two divisors that are two-digit numbers.

   • Analysis: Divisors: $1, 7, 17, 119, 289, 2023$. Two-digit divisors: $17, 119$. True (P).

Zadanie 10

Problem: Compare powers $4^{12}$ and $2^{12}$.

1. Statement: Half of $4^{12}$ equals $2^{12}$.

   • Analysis: $4^{12} = (2^2)^{12} = 2^{24}$. Half: $\frac{2^{24}}{2} = 2^{23}$, not $2^{12}$. False (F).

2. Statement: Twice $2^{23}$ is larger than $2^{24}$.

   • Analysis: $2 \cdot 2^{23} = 2^{24}$, not larger. False (F).

Zadanie 11

Problem: Decide divisibility properties of $N = 10^{12} + 24$.

1. Statement $A$: Divisible by 3.

   • Analysis: Sum of digits of $N$: $1 + 0 + 0 + 0 + \ldots + 0 + 2 + 4 = 7$, not divisible by 3. False (B).

2. Statement $B$: Divisible by 4.

   • Analysis: Last two digits of $N$: $24$, divisible by 4. True (A).

Zadanie 12

Problem: Analyze $n = 5^{40} + 1$, $m = 5^{40} + 3$.

1. Statement $A$: $nm$ divisible by 8.

   • Analysis: $n \mod 8 = (5^{40} + 1) \mod 8 = 2$, $m \mod 8 = (5^{40} + 3) \mod 8 = 4$. $nm = 2 \cdot 4 = 8$, divisible. True (A).

2. Statement $B$: $nm$ divisible by 5.

   • Analysis: Both $n$ and $m$ are not divisible by 5. False (B).

Zadanie 13

Problem: Compare powers $s = 0.2^{0.3}$, $t = 0.2^{0.4}$.

1. Statement $s < t$:

   • Analysis: Since $0.2^{x}$ decreases as $x$ increases, $0.2^{0.3} > 0.2^{0.4}$. False (B).

2. Statement $s > t$:

   • Analysis: Verified above. True (A).

Let me know if you want further details or have specific questions!