Prove that amongst numbers written only using the number 1, i.e.: 1, 11, 111, etc, there is a number that is divisible by 1987.
Prove that there is a power of \(3\) that ends in \(001\). You can take the following fact as given: if the product \(a\times b\) of two numbers is divisible by another number \(c\), but \(a\) and \(c\) share no prime factors (we say that \(a\) and \(c\) are coprime) then \(b\) must be divisible by \(c\).
Prove that in any group of \(10\) whole numbers you can always find some of them that add up to a multiple of \(10\).
Let \(p\) be a prime number, and \(a\) an integer number not divisible by \(p\). Prove that there is a positive integer \(b\) such that \(ab \equiv 1 \pmod p\).
Find the last digit of the number \(1 \times 2 + 2 \times 3 + \dots + 999 \times 1000\).
Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 1989.
Prove that for any number \(d\), which is not divisible by \(2\) or by \(5\), there is a number whose decimal notation contains only ones and which is divisible by \(d\).
\(2n\) diplomats sit around a round table. After a break the same \(2n\) diplomats sit around the same table, but this time in a different order.
Prove that there will always be two diplomats with the same number of people sitting between them, both before and after the break.
Prove that if \(a, b, c\) are odd numbers, then at least one of the numbers \(ab-1\), \(bc-1\), \(ca-1\) is divisible by 4.
10 natural numbers are written on a blackboard. Prove that it is always possible to choose some of these numbers and write “\(+\)” or “\(-\)” between them so that the resulting algebraic sum is divisible by 1001.