Let $x \geqslant 3$. Show that
$$\pi ( x ) \geqslant \frac { \ln ( 2 ) } { 6 } \frac { x } { \ln ( x ) }$$
One may set $n = \lfloor x / 2 \rfloor$ and use Q12.

Let $r \in \mathbb { N } ^ { \star }$. Let $a _ { 1 } , \ldots , a _ { r }$ be non-zero natural integers. Justify that there exists a unique natural integer $d \left( a _ { 1 } , \ldots , a _ { r } \right)$ such that
$$a _ { 1 } \mathbb { Z } \cap a _ { 2 } \mathbb { Z } \cap \cdots \cap a _ { r } \mathbb { Z } = d \left( a _ { 1 } , \ldots , a _ { r } \right) \mathbb { Z }$$

Let $r \in \mathbb { N } ^ { * }$. Let $a _ { 1 } , \ldots , a _ { r }$ be non-zero natural integers. Show that $d \left( a _ { 1 } , \ldots , a _ { r } \right)$ is the smallest non-zero natural integer that is divisible by $a _ { 1 } , \ldots , a _ { r }$.

For all $n \in \mathbb { N } ^ { * }$, we denote by $d _ { n }$ the LCM of the natural integers between 1 and $n$, in other words: $d _ { n } = \operatorname { LCM } ( 1,2 , \ldots , n )$. For all prime number $p$, we denote by $k _ { p }$ the largest natural integer such that $p ^ { k _ { p } } \leqslant n$.
Show that $d _ { n } = \prod _ { \substack { p \leqslant n \\ p \text { prime } } } p ^ { k _ { p } }$.

For all $n \in \mathbb { N } ^ { * }$, we denote by $d _ { n }$ the LCM of the natural integers between 1 and $n$, in other words: $d _ { n } = \operatorname { LCM } ( 1,2 , \ldots , n )$. For all prime number $p$, we denote by $k _ { p }$ the largest natural integer such that $p ^ { k _ { p } } \leqslant n$.
For all prime number $p$, show that $k _ { p } = \left\lfloor \frac { \ln ( n ) } { \ln ( p ) } \right\rfloor$. Deduce that $d _ { n } \leqslant n ^ { \pi ( n ) }$.

For all $n \in \mathbb { N } ^ { * }$, we denote by $d _ { n }$ the LCM of the natural integers between 1 and $n$. Deduce that there exists a non-zero natural integer $N$ such that, for all $n \geqslant N , d _ { n } \leqslant 3 ^ { n }$.
One may use the prime number theorem: $\pi ( x ) \underset { x \rightarrow + \infty } { \sim } \frac { x } { \ln ( x ) }$.

Let $\alpha \in \mathbb { R } _ { + }$. We assume that there exist two sequences of non-zero natural integers $\left( p _ { n } \right) _ { n \in \mathbb { N } }$ and $\left( q _ { n } \right) _ { n \in \mathbb { N } }$ such that
$$\lim _ { n \rightarrow + \infty } \frac { p _ { n } } { q _ { n } } = \alpha \quad \text { and } \quad \left| \alpha - \frac { p _ { n } } { q _ { n } } \right| \underset { n \rightarrow + \infty } { = } o \left( \frac { 1 } { q _ { n } } \right)$$
We further assume that for all $n \in \mathbb { N } , \frac { p _ { n } } { q _ { n } } \neq \alpha$.
Show that $\alpha$ is an irrational number.

Can we apply the result of Q20 to these sequences $\left( p _ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ and $\left( q _ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ to conclude about the irrationality of $\zeta ( 2 )$ ?

150. The four-digit number $\overline{aabb}$ is a perfect square. The remainder when the two-digit number $\overline{ab}$ is divided by 13 is which of the following?
(1) $9$ (2) $10$ (3) $11$ (4) $12$

152. For how many natural numbers less than $50$, is $7^n + 42$ divisible by $43$?
(1) $8$ (2) $7$ (3) $8$ (4) $9$

143- We place each of the integers from 1 to 30 on 30 balls and put them in a bag. We draw at least how many balls to be certain that at least two of the drawn numbers have a greatest common divisor greater than 1?
(1) $15$ (2) $11$ (3) $12$ (4) $13$

150- Seven times a six-digit number $\overline{abcabc}$ is a perfect square. What is the largest value of the number $\overline{abc}$?
$$14 \ (1) \hspace{2cm} 15 \ (2) \hspace{2cm} 16 \ (3) \hspace{2cm} 17 \ (4)$$

151- Two natural numbers equal to $N = \overline{abc}$, written in the form $\varphi(a \circ bc)$ with a change of base, give the value $N$. What is the largest value of $N$, at least how many units less than a perfect square?
$$1 \ (1) \hspace{2cm} 2 \ (2) \hspace{2cm} 3 \ (3) \hspace{2cm} 4 \ (4)$$

152- For how many natural numbers $n$, are the two numbers in the forms $5n-2$ and $7n+3$ not coprime?
$$3 \ (1) \hspace{2cm} 4 \ (2) \hspace{2cm} 5 \ (3) \hspace{2cm} 6 \ (4)$$

144- If $S$ is a subset of natural numbers with 115 elements, when dividing the elements of $S$ by 27, at least how many elements certainly have the same remainder?

(1) $4$

(2) $5$

(3) $6$

(4) $7$

150- How many three-digit numbers exist that are multiples of 11 and whose remainders when divided by both 4 and 5 equal 1?

(1) $3$

(2) $4$

(3) $5$

(4) $6$

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150- The five-digit number $N = \overline{a7\!4\!6b}$ is a multiple of 36. What is the largest remainder when $N$ is divided by 11?
(1) $1$ (2) $2$ (3) $3$ (4) $4$

151- When natural number $A$ is divided by 23, the remainder is 5, and when $A$ is divided by 17, the remainder is 9. What is the largest remainder when the three-digit number $A$ is divided by 12?
(1) zero (2) $2$ (3) $6$ (4) $7$

142. If a number leaves remainders 6 and 11 when divided by 5 and 7 respectively, then when divided by 66, what is the remainder?
(1) $29$ (2) $32$ (3) $40$ (4) $41$

143. For some values of $n \in \mathbb{N}$, if $3 \mid 13n + 3$ and $7 \mid n + 4$ and $\alpha \neq 1$, then what is the smallest sum of digits of $n$?
(1) $7$ (2) $8$ (3) $9$ (4) $10$

145. If $a + 7^{13}$ is divisible by 23, what is the smallest natural number $a$?
(1) $2$ (2) $3$ (3) $4$ (4) $5$

149 -- How many natural multiples of 9 exist such that when divided by 430, the remainder equals the integer part of the quotient?

• [(1)] $4$
• [(2)] $5$
• [(3)] $6$
• [(4)] $7$

150 -- The least common multiple of two numbers is 60 and the greatest common divisor of them is 6. If the sum of these two numbers is 136, what is their difference?

• [(1)] $42$
• [(2)] $48$
• [(3)] $52$
• [(4)] $56$

151 -- If the number $2^n - 1$ is divisible by 217, how many two-digit values does $n$ have?

• [(1)] $4$
• [(2)] $5$
• [(3)] $6$
• [(4)] $7$

131- How many five-digit multiples of 18 are perfect squares? $(\sqrt{10} \cong 3.16)$
(1) $35$ (2) $36$ (3) $37$ (4) $38$