UFM Additional Further Pure

grandes-ecoles 2025 Q13 Bounding or Estimation Proof View

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.

grandes-ecoles 2025 Q14 GCD, LCM, and Coprimality View

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
$$a _ { 1 } \mathbb { Z } \cap a _ { 2 } \mathbb { Z } \cap \cdots \cap a _ { r } \mathbb { Z } = d \left( a _ { 1 } , \ldots , a _ { r } \right) \mathbb { Z }$$

grandes-ecoles 2025 Q15 GCD, LCM, and Coprimality View

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 $a _ { 1 } , \ldots , a _ { r }$.

grandes-ecoles 2025 Q17 GCD, LCM, and Coprimality View

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 $d _ { n } = \prod _ { \substack { p \leqslant n \\ p \text { prime } } } p ^ { k _ { p } }$.

grandes-ecoles 2025 Q18 GCD, LCM, and Coprimality View

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 $d _ { n } \leqslant n ^ { \pi ( n ) }$.

grandes-ecoles 2025 Q19 Prime Counting and Distribution View

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: $\pi ( x ) \underset { x \rightarrow + \infty } { \sim } \frac { x } { \ln ( x ) }$.

grandes-ecoles 2025 Q20 Irrationality and Transcendence Proofs View

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.

grandes-ecoles 2025 Q23 Irrationality and Transcendence Proofs View

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 $\zeta ( 2 )$ ?

iran-konkur 2013 Q150 Quadratic Diophantine Equations and Perfect Squares View

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) $12$

iran-konkur 2013 Q152 Modular Arithmetic Computation View

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

iran-konkur 2014 Q143 GCD, LCM, and Coprimality View

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) $13$

iran-konkur 2014 Q150 Quadratic Diophantine Equations and Perfect Squares View

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

iran-konkur 2014 Q151 Properties of Integer Sequences and Digit Analysis View

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?
$$1 \ (1) \hspace{2cm} 2 \ (2) \hspace{2cm} 3 \ (3) \hspace{2cm} 4 \ (4)$$

iran-konkur 2014 Q152 GCD, LCM, and Coprimality View

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

iran-konkur 2015 Q144 Combinatorial Number Theory and Counting View

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) $7$

iran-konkur 2015 Q150 Modular Arithmetic Computation View

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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iran-konkur 2018 Q150 Divisibility and Divisor Analysis View

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) $4$

iran-konkur 2018 Q151 Modular Arithmetic Computation View

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) $7$

iran-konkur 2019 Q142 Modular Arithmetic Computation View

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) $41$

iran-konkur 2019 Q143 Congruence Reasoning and Parity Arguments View

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) $10$

iran-konkur 2019 Q145 Modular Arithmetic Computation View

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

iran-konkur 2020 Q149 Combinatorial Number Theory and Counting View

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)] $7$

iran-konkur 2020 Q150 GCD, LCM, and Coprimality View

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)] $56$

iran-konkur 2020 Q151 Modular Arithmetic Computation View

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)] $7$

iran-konkur 2021 Q131 Quadratic Diophantine Equations and Perfect Squares View

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

UFM Additional Further Pure

Sequences and Series 205

Number Theory 286

Vector Product and Surfaces 3

Groups 128

Reduction Formulae 4