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 \(1995 \times ***\), which is the number in the fifth row. This is an example of a “long multiplication table”.

Arrange brackets and arithmetic signs around these numbers so that the correct equality is obtained: \[\frac{1}{2}\quad \frac{1}{6}\quad \frac{1}{6009} \ = \ 2003.\]

At the end of the term, Billy wrote out his current singing marks in a row and put a multiplication sign between some of them. The product of the resulting numbers turned out to be equal to 2007. What is Billy’s term mark for singing? (The marks that he can get are between 2 and 5, where 5 is the highest mark).

The best student in the class, Katie, and the second-best, Mike, tried to find the minimum 5-digit number which consists of different even numbers. Katie found her number correctly, but Mike was mistaken. However, it turned out that the difference between Katie and Mike’s numbers was less than 100. What are Katie and Mike’s numbers?

A three-digit number \(ABB\) is given, the product of the digits of which is a two-digit number \(AC\) and the product of the digits of this number is \(C\) (here, as in mathematical puzzles, the digits in the numbers are replaced by letters where the same letters correspond to the same digits and different letters to different digits). Determine the original number.

A girl chose a 4-letter word and replaced each letter with the corresponding number in the alphabet. The number turned out to be 2091425. What word did she choose?

Using five twos, arithmetic operations and exponentiation, form the numbers from 1 to 26.

Using five threes, arithmetic operations and exponentiation, form the numbers from 1 to 39.

Using five fours, arithmetic operations and exponentiation, form the numbers from 1 to 22.

Using five fives, arithmetic operations and exponentiation, form the numbers from 1 to 17.