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: 387

1955 Moscow Mathematical Olympiad, 292
Tags: number theory , divide
Let $a, b, n$ be positive integers, $b < 10$ and $2^n = 10a + b$. Prove that if $n > 3$, then $6$ divides $ab$.

2009 Cuba MO, 9
Tags: number theory , divide
Find all the triples of prime numbers $(p, q, r)$ such that $$p | 2qr + r \,\,\,, \,\,\,q |2pr + p \,\,\, and \,\,\, r | 2pq + q.$$

2011 Belarus Team Selection Test, 2
Tags: number theory , divide , divisible
Do they exist natural numbers $m,x,y$ such that $$m^2 +25 \vdots (2011^x-1007^y) ?$$ S. Finskii

2008 Postal Coaching, 2
Tags: number theory , prime , prime number , divide
Prove that an integer $n \ge 2$ is a prime if and only if $\phi (n)$ divides $(n - 1)$ and $(n + 1)$ divides $\sigma (n)$. [Here $\phi$ is the Totient function and $\sigma $ is the divisor - sum function.] [hide=Hint]$n$ is squarefree[/hide]

2013 Thailand Mathematical Olympiad, 1
Tags: divide , number theory , prime
Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$

2008 Postal Coaching, 3
Tags: number theory , product , divide , divisible
Prove that there exists an in nite sequence $<a_n>$ of positive integers such that for each $k \ge 1$ $(a_1 - 1)(a_2 - 1)(a_3 -1)...(a_k - 1)$ divides $a_1a_2a_3 ...a_k + 1$.

2017 India PRMO, 1
Tags: number theory , sum of digits , divide , divisible
How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?

2018 Puerto Rico Team Selection Test, 3
Tags: divide , number theory , divisible
Let $A$ be a set of $m$ positive integers where $m\ge 1$. Show that there exists a nonempty subset $B$ of $A$ such that the sum of all the elements of $B$ is divisible by $m$.

1965 Dutch Mathematical Olympiad, 2
Tags: perfect square , divide , divisible , number theory
Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$. Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square. Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.

1963 German National Olympiad, 1
Tags: divide , number theory , remainder
a) Prove that when you divide any prime number by $30$, the remainder is either $1$ or is a prime number! b) Does this also apply when dividing a prime number by $60$? Justify your answer!

1989 Tournament Of Towns, (215) 3
Tags: number theory , divide , divisible
Find six distinct positive integers such that the product of any two of them is divisible by their sum. (D. Fomin, Leningrad)

2015 Switzerland - Final Round, 9
Tags: number theory , divide , divisible
Let$ p$ be an odd prime number. Determine the number of tuples $(a_1, a_2, . . . , a_p)$ of natural numbers with the following properties: 1) $1 \le ai \le p$ for all $i = 1, . . . , p$. 2) $a_1 + a_2 + · · · + a_p$ is not divisible by $p$. 3) $a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1$ is divisible by $p$.

2005 iTest, 25
Tags: number theory , divide , divisible
Consider the set $\{1!, 2!, 3!, 4!, …, 2004!, 2005!\}$. How many elements of this set are divisible by $2005$?

2012 Brazil Team Selection Test, 4
Tags: divide , divisible , number theory
Let $p$ be a prime greater than $2$. Prove that there is a prime $q < p$ such that $q^{p-1} - 1$ is not divisible by $p^2$

VMEO III 2006, 10.2
Tags: consecutive , dividisible , divide , number theory
Prove that among $39$ consecutive natural numbers, there is always a number that has sum of its digits divisible by $ 12$. Is it true if we replace $39$ with $38$?

1986 Tournament Of Towns, (129) 4
Tags: factorial , number theory , divisible , divisor , divide
We define $N !!$ to be $N(N - 2)(N -4)...5 \cdot 3 \cdot 1$ if $N$ is odd and $N(N -2)(N -4)... 6\cdot 4\cdot 2$ if $N$ is even . For example, $8 !! = 8 \cdot 6\cdot 4\cdot 2$ , and $9 !! = 9v 7 \cdot 5\cdot 3 \cdot 1$ . Prove that $1986 !! + 1985 !!$ i s divisible by $1987$. (V.V . Proizvolov , Moscow)

1939 Eotvos Mathematical Competition, 2
Tags: number theory , factorial , power of 2 , divide
Determine the highest power of $2$ that divides $2^n!$.

2017 Latvia Baltic Way TST, 16
Tags: combinatorics , number theory , divide , divisible
Strings $a_1, a_2, ... , a_{2016}$ and $b_1, b_2, ... , b_{2016}$ each contain all natural numbers from $1$ to $2016$ exactly once each (in other words, they are both permutations of the numbers $1, 2, ..., 2016$). Prove that different indices $i$ and $j$ can be found such that $a_ib_i- a_jb_j$ is divisible by $2017$.

1958 Polish MO Finals, 1
Tags: perfect cube , divide , number theory
Prove that the product of three consecutive natural numbers, the middle of which is the cube of a natural number, is divisible by $ 504 $ .

2000 Singapore MO Open, 2
Tags: divide , divisible , gcd , prime , number theory
Show that $240$ divides all numbers of the form $p^4 - q^4$, where p and q are prime numbers strictly greater than $5$. Show also that $240$ is the greatest common divisor of all numbers of the form $p^4 - q^4$, with $p$ and $q$ prime numbers strictly greater than $5$.

2006 Thailand Mathematical Olympiad, 5
Tags: divide , divisible , number theory
Show that there are coprime positive integers $m$ and $n$ such that $2549 | (25 \cdot 49)^m + 25^n - 2 \cdot 49^n$

2019 Auckland Mathematical Olympiad, 2
Tags: divisible , divide , number theory
Prove that among any $43$ positive integers there exist two $a$ and $b$ such that $a^2 - b^2$ is divisible by $100$.

2019 Paraguay Mathematical Olympiad, 4
Tags: divisible , divide , number theory
Find the largest positive integer $n$ such that $n^2 + 10$ is divisible by $n-5$.

2013 Saudi Arabia GMO TST, 3
Tags: divide , divisible , number theory
Find the largest integer $k$ such that $k$ divides $n^{55} - n$ for all integer $n$.

2023 Chile National Olympiad, 1
Tags: number theory , divide
Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.