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?
Peter recorded an example of an addition on a board, after which he replaced some digits with letters, with the same figures being replaced with the same letters, and different figures with different letters. He did it such that he was left with the sum: \(CROSS + 2011 = START\). Prove that Peter made a mistake.
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.
Prove that there is a power of 3 that ends in 001.
How many six-digit numbers exist, the numbers of which are either all odd or all even?
Prove that the product of any three consecutive natural numbers is divisible by 6.
How many integers are there from 0 to 999999, in the decimal notation of which there are no two identical numbers next to each other?
Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 1989.
Is it possible to find 57 different two digit numbers, such that no sum of any two of them was equal to 100?
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 \(d\).