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.
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?
The sequence of numbers \(a_1, a_2, \dots\) is given by the conditions \(a_1 = 1\), \(a_2 = 143\) and
for all \(n \geq 2\).
Prove that all members of the sequence are integers.
Compare the numbers: \(A=2011\times 20122012\times 201320132013\) and \(B= 2013\times 20112011 \times 201220122012\).
You are given 12 different whole numbers. Prove that it is possible to choose two of these whose difference is divisible by 11.
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 \(c\).
The numbers \(1, 2, \dots , 9\) are divided into three groups. Prove that the product of the numbers in one of the groups will always be no less than 72.
Prove that in any group of \(10\) whole numbers you can always find some of them that add up to a multiple of \(10\).
You are given \(11\) different positive whole numbers that are less than or equal to \(20\). Prove that it is always possible to choose two numbers where one is divisible by the other.