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?
A student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number, which turned out to be exactly seven times their product. Determine these numbers.
The student did not notice the multiplication sign between two seven-digit numbers and wrote one fourteen-digit number, which turned out to be three times bigger than their product. Determine these numbers.
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 the product of any three consecutive natural numbers is divisible by 6.
Prove that: \[a_1 a_2 a_3 \cdots a_{n-1}a_n \times 10^3 \equiv a_{n-1} a_n \times 10^3 \pmod4,\] where \(n\) is a natural number and \(a_i\) for \(i=1,2,\ldots, n\) are the digits of some number.
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?