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What can you say about the values of $n$ that make $7^n+3^n$ a multiple of 10?
Are there other pairs of integers between 1 and 10 which have similar properties?
Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!
Can you explain the strategy for winning this game with any target?
Is there an efficient way to work out how many factors a large number has?