We solve problems

Problems Algebra and arithmetic Number systems Binary number system Set theory and logic Theory of algotithms

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-103791

Problems Algebra and arithmetic Number systems Decimal number system 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-103802

Problems Algebra and arithmetic Number systems Decimal number system Number theory. Divisibility Divisibility of a number. General properties Set theory and logic Set theory Set cardinality. One-to-one correspondence

12–13 2.0

Which five-digit numbers are there more of: ones that are not divisible by 5 or those with neither the first nor the second digit on the left being a five?

Problem #PRU-104075

Problems Algebra and arithmetic Number systems Decimal number system Methods Trial and error

10–12 2.0

The digits of a 3 digit number \(A\) were written in reverse order and this is the number \(B\). Is it possible to find a value of \(A\) such that the sum of \(A\) and \(B\) has only odd numbers as its digits?

Problem #PRU-104079

Problems Algebra and arithmetic Number systems Decimal number system Methods Trial and error

10–12 2.0

Let \(x\) be a 2 digit number. Let \(A\), \(B\) be the first (tens) and second (units) digits of \(x\), respectively. Suppose \(A\) is twice as large as \(B\). If we add the square of \(A\) to \(x\) then we get the square of a certain whole number. Find the value of \(x\).

Problem #PRU-107749

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

13–15 3.0

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

Problem #PRU-107760

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

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-21988

Problems Algebra and arithmetic Number systems Decimal number system Number theory. Divisibility Division with remainders. Arithmetic of remainders Division with remainder Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

Prove that amongst numbers written only using the number 1, i.e.: 1, 11, 111, etc, there is a number than is divisible by 1987.

Problem #PRU-21989

Problems Algebra and arithmetic Number theory. Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Number systems Decimal number system Euler's theorem Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

Prove that there is a power of 3 that ends in 001.

Problem #PRU-30278

Problems Algebra and arithmetic Number systems Binary number system

11–13 2.0

Suppose you have 127 1p coins. How can you distribute them among 7 coin pouches such that you can give out any amount from 1p to 127p without opening the coin pouches?