### Ewa's Eggs

I put eggs into a basket in groups of 7 and noticed that I could easily have divided them into piles of 2, 3, 4, 5 or 6 and always have one left over. How many eggs were in the basket?

### Inclusion Exclusion

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

### Cogs

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:

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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