This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 196

2019 AMC 12/AHSME, 14
Tags: 2019 AMC 12B , 2019 AMC , AMC , AMC 12 , Divisors , 2019 AMC 10B , AMC 10
Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$ $\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$

1999 Korea Junior Math Olympiad, 6
Tags: KJMO , Divisors , number theory
For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4>m$.

2018 Flanders Math Olympiad, 3
Tags: number theory , Divisors , divisor , Sum
Write down $f(n)$ for the greatest odd divisor of $n \in N_0$. (a) Determine $f (n + 1) + f (n + 2) + ... + f(2n)$. (b) Determine $f(1) + f(2) + f(3) + ... + f(2n)$.

2017 South East Mathematical Olympiad, 4
Tags: number theory , Divisors , Calculate
For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set $$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$where $i = 0, 1, 2, 3$. Determine the smallest positive integer $m$ such that $f_0(m) + f_1(m) - f_2(m) - f_3(m) = 2017$.

1992 ITAMO, 3
Tags: Divisors , factorial , sum of divisors , number theory
Prove that for each $n \ge 3$ there exist $n$ distinct positive divisors $d_1,d_2, ...,d_n$ of $n!$ such that $n! = d_1 +d_2 +...+d_n$.

2017 China Team Selection Test, 1
Tags: number theory , Divisors , modular arithmetic
Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$. Show that there exists a natural $N$ such that for all $n \geq N$, one has $f(n) \leq n^{0.01}$.

2002 IMO Shortlist, 3
Tags: algebra , number theory , primes , Divisors , IMO Shortlist
Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

2024 Korea Junior Math Olympiad (First Round), 9.
Tags: Divisors , number theory
Find the number of positive integers that are equal to or equal to 1000 that have exactly 6 divisors that are perfect squares

2001 All-Russian Olympiad, 4
Tags: number theory , relatively prime , Russia , Divisors , MONT
Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.

2022 IFYM, Sozopol, 7
Tags: number theory , Divisors
Let’s note the set of all integers $n>1$ which are not divisible by a square of a prime number. We define the number $f(n)$ as the greatest amount of divisors of $n$ which could be chosen in such way so that for each two chosen $a$ and $b$, not necessarily different, the number $a^2+ab+b^2+n$ is not a square. Find all $m$ for which there exists $n$ so that $f(n)=m$.

1990 Swedish Mathematical Competition, 1
Tags: Sum , number theory , Divisors
Let $d_1, d_2, ... , d_k$ be the positive divisors of $n = 1990!$. Show that $\sum \frac{d_i}{\sqrt{n}} = \sum \frac{\sqrt{n}}{d_i}$.

2012 Peru MO (ONEM), 1
Tags: Divisors , number theory
For each positive integer $n$ whose canonical decomposition is $n = p_1^{a_1} \cdot p_2^{a_2} \cdot\cdot\cdot p_k^{a_k}$, we define $t(n) = (p_1 + 1) \cdot (p_2 + 1) \cdot\cdot\cdot (p_k + 1)$. For example, $t(20) = t(2^2\cdot 5^1) = (2 + 1) (5 + 1) = 18$, $t(30) = t(2^1\cdot 3^1\cdot 5^1) = (2 + 1) (3 + 1) (5 + 1) = 72$ and $t(125) = t(5^3) = (5 + 1) = 6$ . We say that a positive integer $n$ is [i]special [/i]if $t(n)$ is a divisor of $n$. How many positive divisors of the number $54610$ are special?

1980 Dutch Mathematical Olympiad, 2
Tags: number theory , Divisors
Find the product of all divisors of $1980^n$, $n \ge 1$.

2018 IFYM, Sozopol, 2
Tags: number theory , Divisors , greatest common divisor
$n > 1$ is an odd number and $a_1, a_2, . . . , a_n$ are positive integers such that $gcd(a_1, a_2, . . . , a_n) = 1$. If $d = gcd (a_1^n + a_1.a_2. . . a_n, a_2^n + a_1.a_2. . . a_n, . . . , a_n^n + a_1.a_2. . . a_n) $ find all possible values of $d$.

2013 Saudi Arabia IMO TST, 3
Tags: number theory , Digit , Divisors
For a positive integer $n$, we consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. (For example $n = 117$, has six divisors, namely $1,3,9,13,39,117$. Two of these divisors namely $3$ and $13$, have unit digits equal to $3$. Hence for $n = 117$, $p =33.33...$). Find, when $n$ is any positive integer, the maximum possible value of $p$.

1989 Mexico National Olympiad, 2
Tags: number theory , Divisors
Find two positive integers $a,b$ such that $a | b^2, b^2 | a^3, a^3 | b^4, b^4 | a^5$, but $a^5$ does not divide $b^6$

The Golden Digits 2024, P1
Tags: number theory , Divisors
Let $k\geqslant 2$ be a positive integer and $n>1$ be a composite integer. Let $d_1<\cdots<d_m$ be all the positive divisors of $n{}.$ Is it possible for $d_i+d_{i+1}$ to be a perfect $k$-th power, for every $1\leqslant i<m$? [i]Proposed by Pavel Ciurea[/i]

2019 239 Open Mathematical Olympiad, 2
Tags: number theory , Divisors , number of divisors
Is it true that there are $130$ consecutive natural numbers, such that each of them has exactly $900$ natural divisors?

2005 USAMO, 1
Tags: induction , number theory , prime numbers , Arrangements , relatively prime , Divisors , combinatorics
Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.

2024 Romania EGMO TST, P4
Tags: number theory , Divisors
Find all composite positive integers $a{}$ for which there exists a positive integer $b\geqslant a$ with the same number of divisors as $a{}$ with the following property: if $a_1<\cdots<a_n$ and $b_1<\cdots<b_n$ are the proper divisors of $a{}$ and $b{}$ respectively, then $a_i+b_i, 1\leqslant i\leqslant n$ are the proper divisors of some positive integer $c.{}$

2018 Danube Mathematical Competition, 2
Tags: number theory , Divisors , infinitely many
Prove that there are in finitely many pairs of positive integers $(m, n)$ such that simultaneously $m$ divides $n^2 + 1$ and $n$ divides $m^2 + 1$.

1998 Estonia National Olympiad, 1
Tags: number theory , greatest common divisor , Divisors
Let $d_1$ and $d_2$ be divisors of a positive integer $n$. Suppose that the greatest common divisor of $d_1$ and $n/d_2$ and the greatest common divisor of $d_2$ and $n/d_1$ are equal. Show that $d_1 = d_2$.

2003 France Team Selection Test, 3
Tags: algebra , number theory , primes , Divisors , IMO Shortlist
Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

2007 Thailand Mathematical Olympiad, 16
Tags: number theory , Divisors
What is the smallest positive integer with $24$ positive divisors?

IV Soros Olympiad 1997 - 98 (Russia), 11.2
Tags: number theory , Divisors
Find the three-digit number that has the greatest number of different divisors.