What is the maximum number of consecutive five-digit numbers that have the following property: each number cannot be represented as the product of 2 three-digit numbers?

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An electronic clock shows the time in a standard format (e.g., 20:27). Find the largest possible value of the product of digits on such a clock.

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Replace the asterisks in the number 13**4* with digits so that the number is divisible by 396. Specify all possible variations.

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Find the smallest number whose record consists only of zeros and twos that is divisible by 1125.

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What is the greatest natural degree of the number 2023 that is a divisor of 2023! (factorial)?

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How many ways can you put 7 coins of (a) different denominations into 3 piggy banks? What if the coins are (b) the same? The piggy banks may remain empty.

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Twelve athletes participated in the sprint race. How many ways can medals be distributed if two athletes cannot take the same place?

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There are 20 computers on the same network. From each computer, files were sent to 10 others. Prove that at least 10 pairs of computers exchanged files with each other.

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Find the number of ways to represent the number 5³·7⁴·11⁶ as a product of three positive integers a·b·c.

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The 15 members of the choir must perform at the concert. To do this, they were given concert costumes of three colors: blue, yellow, and green. There are five costumes of each color. To perform, the singers had to line up in a row. The choir director does not like it when 2 people in yellow stand next to each other. How many ways are there to line up the singers so that the choir director likes it?

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A tennis player plays at least 1 game per day and no more than 12 games per week. A week is any 7 consecutive days.

Prove that there is a period of time (counted in full days) in which he will play exactly 20 games.

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Find all prime numbers for which the square of this number increased by 4 and the square of this number increased by 6 are also prime numbers.

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Find all prime numbers p for which the following holds: p = a + b = c - d, where a, b, c, d are prime numbers (not necessarily distinct).

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Find all prime p such that p = a + b, where a, b are composite.

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Is there such a number p that all three numbers p-2023, p, p+2023 are prime?

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Find all pairs of prime numbers p and q such that (p + q)² - pq is a complete square.

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From which of the largest number of coins will it be possible to find a fake coin, which is lighter than the rest, in 4 weighings?

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There are 30 real coins and 31 counterfeit coins. A real coin is 3 grams heavier than a counterfeit coin. The magician chooses a random coin and then determines in one weighing on a scale with an arrow whether it is real or not. How does he do this?

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Four coins and a cup scale are given. It is known that among them there is 1 counterfeit coin, which somehow differs in weight from the real one, but looks the same. What is the least amount of weighing that can be done to determine it?

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There are 60 identical coins, one of which is a counterfeit of a different weight. How can you tell if a counterfeit coin is lighter or heavier in two weighings on a cup scale?

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