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Inclusion Exclusion

How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?

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A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the cogwheel A as the wheels rotate.

School Dinners

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

Every second meal I will have soup for my starter.
Every $5$th meal I will have meat for my main course.
Every $3$rd meal I will have crumble for my dessert.

$2, 5$, and $3$ have no common factors, so I will have eaten $2\times5\times3=30$ school dinners before I sit down to the same combination.

This problem is taken from the UKMT Mathematical Challenges.
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