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### Schools Mathematics Grand Challenge

PROBLEM 4

John does not own a calculator and usually sends a letter to his friend Mary asking for help when he has difficult calculations to do. Recently, Mary received a letter from John asking her to multiply a large number by 750. Unfortunately, when the letter arrived, two of the digits in one of the numbers were unreadable. Mary phoned John to ask what these digits were, but John had forgotten the digits. However, he was sure that both of the missing digits were multiples of 4 (so either 0, 4 or 8). This left Mary with the problem of multiplying the incomplete number:

928X8Y8076726864605652484440363228

(where X and Y are used to denote the missing digits) by the number

750

Mary phoned John again and told him that, while she couldn't do the calculation, she was able to figure out the middle digit of the answer.

What was the middle digit of the product?

PROBLEM 5

The drawer in the locker beside John's bed contains 9 blue socks and 6 black socks. John chooses his socks in the following way: first he takes two socks from the drawer one after the other, and if they match he wears them; otherwise he throws away the first sock drawn out, keeps the second one and takes another sock from the drawer. Once again, if the two socks match (the second and third socks drawn from the drawer) he wears them; otherwise, he throws away the sock that was picked earlier (the second sock) and selects another sock from the drawer. John continues in this manner, always throwing away the sock picked earlier if a pair does not match and then taking another sock from the drawer.

If John never looks at a sock before taking it from the drawer, what is the largest possible number of socks he might have to throw away before he has a matching pair?

PROBLEM 6

You need to know your usercode to solve this puzzle. You should also read the tip of the week to find out about hexadecimal.

Your usercode is a hexadecimal number. To get the answer to this week's question, convert your usercode into decimal, then figure out the remainder when you divide by 13. Give your answer in decimal!