Practice library
Problems
1Find every prime \(p\) for which the number \(p^2 + 11\) has fewer than \(11\) positive divisors.Open2Determine every pair of coprime natural numbers \(m\) and \(n\) for which \[ \frac{5m - n}{m + n} \] is itself a natural number. (Here the natural numbers are the positive integers \(1, 2, 3, \ldots\).) …3Find all pairs of natural numbers \(m\) and \(n\) whose sum equals \(2007\) and whose product is divisible by \(2007\).4Find the smallest natural number \(n\) that is divisible by \(20\) and for which \(n^2\) is a perfect cube and \(n^3\) is a perfect square.5Let \(m\) and \(n\) be positive integers such that \(5m+n\) divides \(5n+m\). Prove that \(m\) divides \(n\).6A kangaroo called Pythagoras likes exactly those natural numbers that are divisible by \(4\), have digit sum \(3\), and have exactly five digits equal to \(0\) in their decimal representation. How many …7Find all prime numbers \(p\), \(q\) and \(r\) that satisfy \[ r^{4} = pq + 4 . \]8Find all natural numbers \(n\) whose cube equals the sum of the squares of three divisors of \(n\), where the three divisors need not be different from one another.9Prove that there are no natural numbers \(a\) and \(b\) satisfying \[ \sqrt{a} + \sqrt{b} = \sqrt{2021} . \]10Find all pairs of natural numbers \(a\) and \(b\) for which \[ v = ab - 2a - 4b , \] where \(v\) denotes the least common multiple of \(a\) and \(b\).11Determine the smallest possible value of \[ \left|25^{m} - 36^{n}\right| \] where \(m\) and \(n\) are positive integers.12Ana picked the eight digits \(1, 2, 3, 4, 5, 6, 7\) and \(9\). She then forms groups of four two-digit primes, each group using all of her chosen digits. What is the sum of the four primes in one such …13Find all pairs of coprime integers \(x\) and \(y\) that satisfy the equation \[ 4x^{3} + y^{3} = 3xy^{2} . \]14Determine all quintuples of primes \(p_{1} \le p_{2} \le p_{3} \le p_{4} \le p_{5}\) with the property that each of the five primes divides the sum of the remaining four.15A teacher handed Matej four sheets of paper, each carrying one nonzero digit. Matej laid the sheets in a row and so formed a four-digit number. He then interchanged two of the sheets, without flipping …16We want to choose a set \(P\) of \(k\) primes and a set \(N\) of \(n\) consecutive positive integers in such a way that every number \(a \in N\) is divisible by at least one prime \(p \in P\). (a) Determine …
Showing 16 of 651 - problem statements are free for everyone.