2016 digits are written in a circle. It is known, that if you make a number reading the digits clockwise, starting from some particular place, then the resulting 2016-digit number is divisible by 27. Show that if you start from some other place, and moving clockwise make up another 2016-digit number, then this new number is also divisible by 27.
We call a \(10\)-digit number interesting if it is divisible by \(11111\), and all its digits are different. How many interesting numbers does there exist?
Note that a number \(k = a_0 + 10a_1 + \dots +10^9 a_9\) is divisible by \(11111\) if and only if a number \(m = (a_0+a_5) +10(a_1+a_6) + \dots + 10^4 (a_4+a_9)\) is also divisible by \(11111\). This is because \(100000=1+9 \times 11111\) and we subtract \(99999 (a_5 + 10a_6 + 100a_7 + 1000a_8 +10000a_9)\) from the original number.
Can you pay 20 p in seven coins?
Can you pay 20 p in seven coins using 1p and 5p coins only?
Can you pay 25 p in eight coins using 1p and 5p coins only?
(a) Show that it is impossible to find five odd numbers which all add to 100.
(b) Alice wrote several odd numbers on a piece of paper. The Hatter did not see the numbers, but says that if he knew how many numbers Alice wrote down, than he would say with certainty if the sum of the numbers is even or odd. How can he do it?
At the tea party the Hatter, who loves everything being odd, decided to divide 25 cakes between himself, the March Hare, Alice, and the Dormouse in such a way that everybody receives an odd number of cakes. Show that he would never be able to do it.
Alice went to a shop to buy flowers for her sister. She bought 6 roses £1 each, 4 lilies 82p each, and 4 freesias 76p each. At the till she was asked to pay £12.25, which she asked to recalculate straight away pointing out that the amount was not correct. Alice did not calculate the full amount, but how did she know that they made a mistake?
The Hatter says that he knows four numbers such that their sum and their product are odd. Is he right? Can you expain why?
The four numbers 1, 1, 1, 2 are written on a piece of paper. Alice and the Hatter are playing a game. They add in turns 1 to any two numbers to make the new four numbers. The winner is the one to make all the four numbers equal. If Alice goes first, who will win, if any?