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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

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

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-116474

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

10–11 3.0

Four numbers (from 1 to 9) have been used to create two numbers with four-digits each. These two numbers are the maximum and minimum numbers, respectively, possible. The sum of these two numbers is equal to 11990. What could the two numbers be?

Problem #PRU-21988

Problems Number Theory Numeral systems Decimal number system 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 Number Theory Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Numeral 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-32787

Problems Number Theory Numeral systems Decimal number system Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

12–13 3.0

Is it possible to find 57 different two digit numbers, such that no sum of any two of them was equal to 100?

Problem #PRU-60739

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Numeral systems Decimal number system Methods Pigeonhole principle Pigeonhole principle (other)

13–16 3.0

Prove that for any natural number there is a multiple of it, the decimal notation of which consists of only 0 and 1.

Problem #PRU-60806

Problems Number Theory Numeral systems Decimal number system Divisibility Divisibility rules Divisibility tests for 3 and 9 Methods Pigeonhole principle Pigeonhole principle (other)

13–15 3.0

Without calculating the answer to \(2^{30}\), prove that it contains at least two identical digits.

Problem #PRU-78508

Problems Number Theory Numeral systems Decimal number system Methods Pigeonhole principle Pigeonhole principle (other)

14–16 3.0

Prove that in a group of 11 arbitrary infinitely long decimal numbers, it is possible to choose two whose difference contains either, in decimal form, an infinite number of zeroes or an infinite number of nines.