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?
One three-digit number consists of different digits that are in ascending order, and in its name all words begin with the same letter. The other three-digit number, on the contrary, consists of identical digits, but in its name all words begin with different letters. What are these numbers?
Write the first 10 prime numbers in a line. How can you remove 6 digits to get the largest possible number?
In one move, it is permitted to either double a number or to erase its last digit. Is it possible to get the number 14 from the number 458 in a few moves?
Find the largest six-digit number, for which each digit, starting with the third, is equal to the sum of the two previous digits.
Find the largest number of which each digit, starting with the third, is equal to the sum of the two previous digits.
Find a two-digit number that is 5 times the sum of its digits.
When we write \(137\) in decimal, we mean \(1 \times 10^2 + 3 \times 10 + 7 \times 1\). If we write using powers of \(2\) instead of powers of \(10\), we have \(137 = 1 \times 2^7 + 0 \times 2^6 + 0 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0\). This is called its binary representation. To tell apart binary representation from decimals, we can use the following notation: \(137 = (10001001)_2\).
What is the number \(273\) in binary? In the next few problems, we will see that using the binary representation of a number is a very useful tool to finding whether a particular Nim game is a winning position or a losing position.
Are there any two-digit numbers which are the product of their digits?
A teacher saw the calculation \(3\times 4 = 10\) written on the whiteboard. She was about to erase it, thinking it was wrong, but then wondered whether it might have been written in a different numeral system.
Is it possible that this multiplication is correct in some base? If so, which one?