Math Problems and solutions
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12.2.29 Number theory
0 $
12.2.30 Number theory
2.57 $
12.2.31 Number theory
12.2.32 Number theory
3.08 $
12.2.33 Number theory
3.85 $
12.2.34 Number theory
12.2.35 Number theory
3.34 $
12.2.36 Number theory
7.7 $
12.2.37 Number theory
5.13 $
12.2.38 Number theory
6.42 $
12.2.39 Number theory
12.2.40 Number theory
12.2.41 Number theory
12.2.42 Number theory
1.28 $
19.6.1.5 Lebesgue measure and integration
![Problem: Let \( a \) be such an integer, that \[ 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{23}=\frac{a}{23 !} \text {. } \] Calculate the remainder from dividing \( a \) by 13 .](/storage/uploads/task_mls/1704/en/12.2.37.png){kind=link}
![Problem: Let \( p \) is prime, greater than 3 and \( m \) and \( n \) are such coprime integers that \[ \frac{m}{n}=\frac{1}{1^{2}}+\frac{1}{2^{2}}+\cdots+\frac{1}{(p-1)^{2}} . \] Let's prove that \( m \) is divided by \( p \).](/storage/uploads/task_mls/1705/en/12.2.38.png){kind=link}
![Problem: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \). Prove that \[ \{x \in \mathbb{R}: f(x) \geq c\}=\bigcap_{n=1}^{\infty}\left\{x \in \mathbb{R}: f(x) \geq c-\frac{1}{n}\right\} . \]](/storage/uploads/task_mls/1710/en/19.6.1.5.png){kind=link}