We solve problems

Problem #PRU-103964

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

There are \(n\) integers. Prove that among them either there are several numbers whose sum is divisible by \(n\) or there is one number divisible by \(n\) itself.

Problem #PRU-30375

Problems Number Theory Divisibility

12–14 2.0

Prove that \(n^2 + 1\) is not divisible by \(3\) for any natural \(n\).

Problem #PRU-31304

Problems Number Theory Divisibility Equations in integer numbers

12–14 3.0

Solve the equation with natural numbers \(1 + m + m^2 + m^3 = 2^n\).

Problem #PRU-102997

Problems Number Theory Numeral systems Binary number system Discrete Mathematics Algorithm Theory

10–12 3.0

Michael thinks of a number no less than \(1\) and no greater than \(1000\). Victoria is only allowed to ask questions to which Michael can answer “yes” or “no” (Michael always tells the truth). Can Victoria figure out which number Michael thought of by asking \(10\) questions?

Problem #PRU-103729

Problems Number Theory Divisibility Odd and even numbers

10–12 2.0

The parliament of a certain country has two houses with an equal number of members. In order to make a decision on an important issue all the members voted and there were no abstentions. When the chairman announced that the decision had been taken with a 23-vote advantage, the opposition leader declared that the results had been rigged. How did he know it?

Problem #PRU-103791

Problems Number Theory Numeral systems Decimal number system Discrete Mathematics Set theory and logic Mathematical logic Puzzles

12–12 3.0

In the rebus below, replace the letters with numbers such that the same numbers are represented with the same letter. The asterisks can be replaced with any numbers such that the equations hold.

An explanation of the notation used: the unknown numbers in the third and fourth rows are the results of multiplying 1995 by each digit of the number in the second row, respectively. These third and fourth rows are added together to get the total result of the multiplication \(1995 \times ***\), which is the number in the fifth row. This is an example of a “long multiplication table”.

Problem #PRU-107749

Problems Number Theory Numeral systems Decimal number system Algebra Algebraic equations and systems of equations Diophantine equations

13–15 3.0

A student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number, which turned out to be exactly seven times their product. Determine these numbers.

Problem #PRU-107760

Problems Number Theory Numeral systems Decimal number system Algebra Algebraic equations and systems of equations Diophantine equations

13–15 3.0

The student did not notice the multiplication sign between two seven-digit numbers and wrote one fourteen-digit number, which turned out to be three times bigger than their product. Determine these numbers.

Problem #PRU-107845

Problems Number Theory Divisibility Equations in integer numbers

11–14 3.0

Is it possible to find natural numbers \(x\), \(y\) and \(z\) which satisfy the equation \(28x+30y+31z=365\)?

Problem #PRU-109603

Problems Number Theory Divisibility Divisibility of a number. General properties Methods Examples and counterexamples. Constructive proofs Calculus Number sequences Number sequences (other)

14–16 5.0

Is there a sequence of natural numbers in which every natural number occurs exactly once, and for any \(k = 1, 2, 3, \dots\) the sum of the first \(k\) terms of the sequence is divisible by \(k\)?