Practice library
Problems
1The value of the expression \(10^{2016} - 10^{15}\) is a natural number. What is the sum of the digits of that number? A \(1\) B \(17\) C \(2001\) D \(18\,000\) E \(18\,009\)Open2The digits of a five-digit number add up to \(44\). What is the product of the digits of that number? A \(2^{3} \cdot 3^{8}\) B \(2^{3} \cdot 9^{3}\) C \(8 \cdot 4^{9}\) D \(8 \cdot 3^{4}\) E None of the …3Let \(x\), \(y\), \(z\) and \(w\) be natural numbers. At most how many of the six sums \[ x+y, \quad x+z, \quad x+w, \quad y+z, \quad y+w, \quad z+w \] can be odd? A \(2\) B \(3\) C \(4\) D \(5\) E \(6\) …4For how many integers \(k\) is the number \(k + 6\) an integer multiple of the number \(k - 6\)? A \(0\) B \(4\) C \(6\) D \(8\) E \(12\)5What is the largest natural number \(n\) with the property that, when \(n\) is divided by \(20\), the remainder is equal to the quotient? A \(21\) B \(92\) C \(231\) D \(399\) E \(440\)6Let \(N\) be the number whose decimal expansion consists of \(2025\) copies of the digit \(3\): \[ N = \underbrace{33\ldots3}_{2025} . \] What remainder does \(N\) leave when it is divided by \(12\)? A …7Let \(k\), \(m\) and \(n\) be natural numbers, none of which is divisible by \(5\). Prove that at least one of the three numbers \(k^2 - m^2, \qquad m^2 - n^2, \qquad n^2 - k^2\) is divisible by \(5\). …8The smallest natural number whose square ends in three fours is \(38\), because \(38^2 = 1444\). Which natural number is the next smallest one with this property?9A prime number \(p\) is given. Find all pairs of natural numbers \(x\) and \(y\) (positive integers) that satisfy \[ p\,(x - 5) = x\,y . \]10Is there a natural number \(n\) with the following property: if \(n\) is multiplied by the sum of its own digits, then the digits of the resulting product add up to \(3\)?11Find all pairs of prime numbers \(p\) and \(q\) for which the number \[ 2p^{2}q + 45pq^{2} \] is a perfect square.12The integers \(a\), \(b\), \(c\), \(d\) satisfy \(a > b > c > d\) and \[ (1 - a)(1 - b)(1 - c)(1 - d) = 10 . \] Which values can the expression \(a + b - c - d\) take?13Determine all prime numbers \(p\) and \(q\) for which the number \[ 2^{2} + p^{2} + q^{2} \] is also prime.14Find every natural number \(n \ge 10\) all of whose decimal digits are nonzero and which has the following property: deleting any single digit of \(n\) leaves a number that divides \(n\).15Find all triples of prime numbers \(p\), \(q\), \(r\) for which \[ p + q^{2} = r^{4} . \]16Eva, Igor, Marko and Marusa each wrote one natural number on a sheet of paper. Deleting the last digit of Eva's number gives Igor's number; deleting the last digit of Igor's number gives Marko's number; …17Tine collects stamps. For his birthday he received a new album with room for plenty of them, so he took \(2002\) tolars out of his money box and decided to spend all of it on stamps. A friend offered him …18Oliver rolled an ordinary die \(100\) times and multiplied together all \(100\) numbers that came up on top. The product he obtained was \(6^{70}\). What is the smallest possible number of rolls on which …19Find all primes \(p\), \(q\), \(r\) and \(s\) for which \[ p + q = r \qquad\text{and}\qquad q + r = s^{2} . \]20A large rectangle is cut into seven smaller rectangles, each of which has both of its side lengths equal to a whole number of metres. Five of the seven pieces have their areas written in them, as shown …21Find every three-digit number \(\overline{abc}\) whose digits \(a\), \(b\), \(c\) are all nonzero and which satisfies \[ \overline{abc} = 3 \cdot a! + 2 \cdot b! + c! . \] Here \(\overline{abc}\) is the …22Find every prime \(p\) for which the number \(p^2 + 11\) has fewer than \(11\) positive divisors.23Determine 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 \(1, 2, 3, \ldots\).) …24Find all pairs of natural numbers \(m\) and \(n\) whose sum equals \(2007\) and whose product is divisible by \(2007\).25Find 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.26Let \(m\) and \(n\) be positive integers such that \(5m+n\) divides \(5n+m\). Prove that \(m\) divides \(n\).27A 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 …28Find all prime numbers \(p\), \(q\) and \(r\) that satisfy \[ r^{4} = pq + 4 . \]29Find 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.30Prove that there are no natural numbers \(a\) and \(b\) satisfying \[ \sqrt{a} + \sqrt{b} = \sqrt{2021} . \]31Find all pairs of natural numbers \(a\) and \(b\) for which \[ v = ab - 2a - 4b , \] where \(v\) denotes the least common multiple of \(a\) and \(b\).32Determine the smallest possible value of \[ \left|25^{m} - 36^{n}\right| \] where \(m\) and \(n\) are positive integers.
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