11 GPFs Positive integers (counting numbers) which have more than two factors are called composite numbers or simply composites. They are not prime but they do have factors which are prime. For example, here are some prime factorisations: 20 = 2 × 2 × 5, Thus the greatest prime factors (GPFs) of 20, 21, 22 are 5, 7, 11 respec- 21 = 3 × 7, 22 = 2 x 11. tively. The list of GPFs of successive composites is called a GPF sequence. For example, the sequence of GPFs for the composites from 40 to 49 is 5, 7, 11, 5, 23, 3, 7. Note that 41, 43, and 47 are prime and do not contribute to the sequence of GPFs. atorget a Find the sequence of GPFs for the composites from 60 to 65. b Explain why successive composites that give the sequence of GPFs 41, 19, 79 must all have at least four digits. c Find the smallest successive composites that give the sequence of GPFs 17, 73, 2, 19. 510, 511,512, 513 d Find the largest composite less than 10 000 with a GPF of 7 and prove that it is the largest. 7, 19, 23, 29, 31, 37,

A student is told to work any 8 out of 15 questions on an exam. In how   many different ways can he complete the exam?

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The fourth door is very tricky. It has a 4-number lock to escape consisting of the numbers from 1 to 36. Here are your clues: a. The first number is the sum of the digits in the year that Gottfried Leibniz published his works on Calculus. c. The third number is the sum of the digits of the year that Newton died. d. The last number is the 5th and 6" decimal digits in order of V57.