On a recent cruise one of our colleagues was using the running track that went around the top deck. There was a sign saying "Four laps is a kilometre, six laps is a mile", which somewhat confused him.
Assuming that the first part is correct, then the track would be 250 metres long, and so six laps would be 1,500 metres. Now, the 1,500 metres race is known as the "Metric Mile", but that does not make it an actual mile. In fact a mile is just under 1,610 metres.
And this set him thinking - what is the longest track where a whole number of laps would be both 1,000 metres (a kilometre) and 1,610 metres (a mile)?
So, how long would the track be?
What you need to find is the highest common factor of 1,000 and 1,610.
Start by finding the prime factors of both numbers:
1000 = 2 x 2 x 2 x 5 x 5 x 5 and 1610 = 2 x 5 x 7 x 23
The only common factors are 2 and 5, so the track would have to be 10 metres long - perhaps this is why the track is the length it is!