Problems

In the rebus below, replace the letters with numbers such that the same numbers are represented with the same letter. The asterisks can be replaced with any numbers such that the equations hold.

An explanation of the notation used: the unknown numbers in the third and fourth rows are the results of multiplying 1995 by each digit of the number in the second row, respectively. These third and fourth rows are added together to get the total result of the multiplication $1995\times \ast \ast \ast$, which is the number in the fifth row. This is an example of a “long multiplication table”.

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.

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?

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 $a_1{x}_1+\cdots +a_n{x}_n$, 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: $$a_1{a}_2{a}_3\cdots a_{n-1}{a}_n\times {10}^3\equiv a_{n-1}{a}_n\times {10}^3(\mathrm{mod}4),$$ where $n$ is a natural number and $a_i$ for $i=1,2,\dots ,n$ are the digits of some number.

Jane wrote a number on the whiteboard. Then, she looked at it and she noticed it lacks her favourite digit: 5. So she wrote 5 at the end of it. She then realized the new number is larger than the original one by exactly 1661. What is the number written on the board?

Replace letters with digits to maximize the expression: $$NO+MORE+MATH$$ (same letters stand for identical digits and different letters stand for different digits.)

The digits of a 3 digit number $A$ were written in reverse order and this is the number $B$. Is it possible to find a value of $A$ such that the sum of $A$ and $B$ has only odd numbers as its digits?

Let $x$ be a 2 digit number. Let $A$, $B$ be the first (tens) and second (units) digits of $x$, respectively. Suppose $A$ is twice as large as $B$. If we add the square of $A$ to $x$ then we get the square of a certain whole number. Find the value of $x$.