Prove that if \(x_0^4 + a_1x_0^3 + a_2x_0^2 + a_3x_0 + a_4\) and \(4x_0^3 + 3a_1x_0^2 + 2a_2x_0 + a_3 = 0\) then \(x^4 + a_1x^3 + a_2x^2 + a_3x + a_4\) is divisible by \((x - x_0)^2\).
Two people play a game with the following rules: one of them guesses a set of integers \((x_1, x_2, \dots , x_n)\) which are single-valued digits and can be either positive or negative. The second person is allowed to ask what is the sum \(a_1x_1 + \dots + a_nx_n\), where \((a_1, \dots ,a_n)\) is any set. What is the smallest number of questions for which the guesser recognizes the intended set?
Does there exist a number \(h\) such that for any natural number \(n\) the number \(\lfloor h \times 2021^n\rfloor\) is not divisible by \(\lfloor h \times 2021^{n-1}\rfloor\)?
What figure should I put in place of the “?” in the number \(888 \dots 88\,?\,99 \dots 999\) (eights and nines are written 50 times each) so that it is divisible by 7?
A numerical sequence is defined by the following conditions: \[a_1 = 1, \quad a_{n+1} = a_n + \lfloor \sqrt{a_n}\rfloor .\]
Prove that among the terms of this sequence there are an infinite number of complete squares.
On a table there are 2022 cards with the numbers 1, 2, 3, ..., 2022. Two players take one card in turn. After all the cards are taken, the winner is the one who has a greater last digit of the sum of the numbers on the cards taken. Find out which of the players can always win regardless of the opponent’s strategy, and also explain how he should go about playing.
Prove the divisibility rule for \(3\): the number is divisible by \(3\) if and only if the sum of its digits is divisible by \(3\).
Find the smallest \(k\) such that \(k!\) (\(k!= k\times(k-1)\times \ldots \times 1\)) is divisible by \(2024\).
While studying numbers and its properites, Robinson came across a 3-digit prime number with the last digit being equal to the sum of the first two digits. What was the last digit of that number if among the number did not have any zeros among it’s digits?
Prove the divisibility rule for \(4\): a number is divisible by \(4\) if and only if the number made by the last two digits of the original number is divisible by \(4\);
Can you come up with a divisibility rule for \(8\)?