The student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number, which turned out to be seven times bigger than 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.
Two people play a game with the following rules: one of them guesses a set of integers \((x_1, x_2, \dots , x_n)\) which are single-valued digits and can be either positive or negative. The second person is allowed to ask what is the sum \(a_1x_1 + \dots + a_nx_n\), where \((a_1, \dots ,a_n)\) is any set. What is the smallest number of questions for which the guesser recognizes the intended set?
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 distinct seven-digit numbers exist? It is assumed that the first digit cannot be zero.
We call a natural number “fancy”, if it is made up only of odd digits. How many four-digit “fancy” numbers are there?
We are given 51 two-digit numbers – we will count one-digit numbers as two-digit numbers with a leading 0. Prove that it is possible to choose 6 of these so that no two of them have the same digit in the same column.
Prove that amongst any 11 different decimal fractions of infinite length, there will be two whose digits in the same column – 10ths, 100s, 1000s, etc – coincide (are the same) an infinite number of times.
How many six-digit numbers exist, for which each succeeding number is smaller than the previous one?
Why are the equalities \(11^2 = 121\) and \(11^3 = 1331\) similar to the lines of Pascal’s triangle? What is \(11^4\) equal to?