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#### Decimal number system

Find a two-digit number that is 5 times the sum of its digits.

#### Decimal number system

Find the largest number of which each digit, starting with the third, is equal to the sum of the two previous digits.

#### Decimal number system

Find the largest six-digit number, for which each digit, starting with the third, is equal to the sum of the two previous digits.

#### Decimal number system , Theory of algorithms (other)

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?

#### Decimal number system , Proof by exhaustion

Write the first 10 prime numbers in a line. How can you remove 6 digits to get the largest possible number?

#### Decimal number system

Try to find all natural numbers which are five times greater than their last digit.

#### Decimal number system , Pigeonhole principle (other)

7 different digits are given. Prove that for any natural number n there is a pair of these digits, the sum of which ends in the same digit as the number.

#### Decimal number system , Pigeonhole principle (other)

What is the maximum difference between neighbouring numbers, whose sum of digits is divisible by 7?

#### Decimal number system , Puzzles

To a certain number, we add the sum of its digits and the answer we get is 2014. Give an example of such a number.

#### Decimal number system , Divisibility tests for 3 and 9 , Pigeonhole principle (other)

Without calculating the answer to $2^{30}$, prove that it contains at least two identical digits.

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