You, of course, know that there are even and odd numbers.
Even numbers are those that are divisible by 2 without a remainder (for example, 2, 4, 6, etc.). Each such number can be written as 2K by choosing a suitable integer K (for example, 4 = 2 x 2, 6 = 2 x 3, etc.).
Odd numbers are those that, when divided by 2, leave a remainder of 1 (for example, 1, 3, 5, etc.). Each such number can be written as 2K + 1 by choosing a suitable integer K (for example, 3 = 2 x 1 + 1, 5 = 2 x 2 + 1, etc.).
Even and odd numbers have remarkable properties:
a) the sum of two even numbers is even;
b) the sum of two odd numbers is even;
c) the sum of even and odd numbers is odd number.
1. Represent each of the numbers 1101 and –1101 as a) 2n + 1; b) 2n – 1; c) 2n + 333.
2 .The product of any two odd numbers is odd, but the sum is even. Prove it.
3. Prove that if the sum of two integers is odd, then the product of these numbers is even.
4. The difference of two integers is multiplied by their product. Could you get this number 11011811061018224521543?
5. Twenty years ago, banknotes in denominations of 1, 3, 5, 10 and 25 rubles were in use. Prove that if 25 rubles were exchanged for ten such banknotes, then at least one of these ten bills is a ten.
6. Ev writes one whole number on the board, and Od writes another. If the product is even, Ev is declared the winner; if it is odd, then Od is declared the winner. Can one of the players play in such a way as to definitely win?
7. 100 chips are placed in a row. It is allowed to swap places of any two chips that stand on the same parity positions. Is it possible to put the chips in reverse order?
8. There are 100 people in the company. Three are on duty every night. Is it possible to organize duty in such a way that after a while everyone will be on duty with everyone once?
9. Numbers from 1 to 10 are written in a row. Is it possible to place “+” and “–” signs between them so that the value of the resulting expression is equal to zero?
10. Is it possible to erase one of the given a) 1992; b) 1993 integers so that the sum of the remaining numbers is even?
11. Is it possible to divide the natural numbers 1, 2, ..., 20, 21 into several groups, in each of which the largest number is equal to the sum of all other numbers in this group?
12. Six numbers are given: 1, 2, 3, 4, 5, 6. You are allowed to add one to any two numbers. Is it possible to make all numbers equal using several such operations?
13. On 99 cards, write the numbers 1, 2, ..., 99, mix them, lay them out with their blank sides up and write the numbers 1, 2, ..., 99 again. For each card, add up its two numbers and multiply the 99 resulting sums. Prove that the result is even.
14. The vertices and centres of the faces are marked on the cube, and the diagonals of all faces are drawn. Is it possible to go around all marked points along diagonal segments, visiting each of them exactly once?
15. There is a king on a certain square of the chessboard. Two people take turns moving it around the board. It is forbidden to return the king to the field where it was just there. The player who places the king on a field where the king has already been will win. Which player can guarantee victory?
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