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**Bartimaeus** Thye underground temple of the ancient primes was recently disovered deep in a South American jungle. The keys to the temple were discovered long ago. There are 25 of them, each one numbered with a different prime less than 100.

Each of the temple's doors has a lock which requires a set of keys to open it. The numbers on the keys which open a lock always adds to 100, the sacred number of the ancient primes.

The keys which open the outer door of the temple are numbered 2, 3, 5, 7, 11, 13, 17, 19, 23. Doors to unimportant rooms inside only need two keys to open them.

a) Show that no door can be unlocked with exactly eight keys.

Please include an argument.

There are exactly 35 selections of seven keys which add to 100. So it is not surprising that the temple's main chamber has 35 entry doors, each needing a different set of seven keys to open it.

b) All of these 35 doors are eventually opened. Find the selection of keys used on one of these doors which contains the key with the highest number.

Please include an argument with the correct set.

c) Find the selection of seven keys whose number have the largest product.

Please include a suitable strategy, correct set and supporting argument.

Thanks again guys! :)