James is playing a card game called sum game. He holds 8 cards, numbered 1 to 8. Each time he places a card on the table he finds the sum of all the cards he has placed on the the table so far. To win the game , alternate sums must be divisible by two different numbers. For example, he plays the cards in order 5, 1, 4, 2, 8, 7, 3, 6, the sums are 5, 6,10, 12, 20, 27, 30, 36 which are alternately divisible by 5 and 3, starting with 5.
1. James now plays with 17 cards, numbered 1 to 17. SHow that he cannont place the cards so that the alternate sums are divisible by 4 and 9, starting with 4
2. James has cards numbered 1 to n. He places them so that the alternate sums are divisible by positive integers a and b, starting with a. Show that, if the card numbered 1 is placed in an even position, the maximum value for a + b is 2n + 1