There are four numbers written in a row. The first number is \(100\). It is known that if we divide the first number by the second number we will get a prime number as a result, if we the second number by the third number we will get a prime number, and if we divide the third number by the fourth number we will also get a prime number. Can all the resulting prime numbers be distinct?
A five-digit number is called indecomposable if it is not decomposed into the product of two three-digit numbers. What is the largest number of indecomposable five-digit numbers that can come in a row?
Let’s "prove" that the number \(1\) is a multiple of \(3\). We will use the symbol \(\equiv\) to denote "congruent modulo \(3\)". Thus, what we need to prove is that \(1\equiv 0\) modulo \(3\). Let’s see: \(1\equiv 4\) modulo \(3\) means that \(2^1\equiv 2^4\) modulo \(3\), thus \(2\equiv 16\) modulo \(3\), however \(16\) gives the remainder \(1\) after division by \(3\), thus we get \(2\equiv 1\) modulo \(3\), next \(2-1\equiv 1-1\) modulo \(3\), and thus \(1\equiv 0\) modulo \(3\). Which means that \(1\) is divisible by \(3\).
The numbers \(a\) and \(b\) are integers and the number \(p \ge 3\) is prime. Suppose that \(a+b\) and \(a^2 +b^2\) are divisible by \(p\). Show that \(a^2 + b^2\) is divisible by \(p^2\).
Find all possible non-zero digits \(A\) for which the following holds \((AA+AA+1) \times A = AAA\). (Recall \(AA\) means the two-digit number whose first and second digits are \(A\))
Show that for any natural number \(n\geq 1\), the number \[n^4+4n^2+9\] is not a perfect square.
You start with the number \(1\) on a piece of paper. You may perform two operations:
double up (multiply the number on your paper by \(2\), and erase the old number), increase by \(1\) (add \(1\) to the number on your paper, and erase the old number).
So for example, you may end up with the numbers \(1,2,3,6,\cdots\). Show that it is possible to obtain \(975\) starting from \(1\) in at most \(16\) operations.
A collection of weights is made from the weights \(1,2,4,8,\dots\) grams (that is, all powers of \(2\)). Some weights may appear several times. The weights are placed on the two pans of a balance scale and the scale is in balance. It is known that all the weights on the left pan are different.
Prove that the number of weights on the right pan is at least as large as the number of weights on the left pan.
Cut a \(7\times 7\) square into \(9\) rectangles, out of which you can construct any rectangle whose sidelengths are less than \(7\). Show how to construct the rectangles.
There are \(2^4\) cities in a kingdom. Show that it is possible to build a system of roads so that one can travel from any city to any other while passing through at most one intermediate city, and so that at most five roads leave each city.