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: 133

1984 AMC 12/AHSME, 9
Tags: floor function , logarithm , prime factorization , number theory
The number of digits in $4^{16} 5^{25}$ (when written in the usual base 10 form) is A. 31 B. 30 C. 29 D. 28 E. 27

2006 Estonia Team Selection Test, 6
Tags: prime number , prime factorization , number theory
Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$. Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$. (b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.

2008 Paraguay Mathematical Olympiad, 1
Tags: prime factorization , number theory
How many positive integers $n < 500$ exist such that its prime factors are exclusively $2$, $7$, $11$, or a combination of these?

1986 IMO Shortlist, 6
Tags: number theory , prime factorization , divisor
Find four positive integers each not exceeding $70000$ and each having more than $100$ divisors.

2019 Malaysia National Olympiad, 6
Tags: prime factorization , algebra
It is known that $2018(2019^{39}+2019^{37}+...+2019)+1$ is prime. How many positive factors does $2019^{41}+1$ have?

2009 AMC 8, 16
Tags: number theory , prime factorization
How many $ 3$-digit positive integers have digits whose product equals $ 24$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 24$

2010 Princeton University Math Competition, 1
Tags: prime factorization , number theory
Find the positive integer less than 18 with the most positive divisors.

2010 Math Prize For Girls Problems, 8
Tags: prime factorization , number theory
When Meena turned 16 years old, her parents gave her a cake with $n$ candles, where $n$ has exactly 16 different positive integer divisors. What is the smallest possible value of $n$?