Math CentralQuandaries & Queries


Question from Bill, a student:

Four friends are racing together down a long flight of stairs. A goes 2 steps at a time, B 3 at a time, C 4 at a time and D 5 and at a time. The only steps on which all four tread are the top one and the bottom one. How many stairs in the flight are stepped on exactly once?

Hi Bill,

How many steps are there? A lands on steps that are multiples of 2, B on steps that are multiples of 3, and similarly for C and D. They are at the top when they all land on the same step for the first time. What is the smallest number that is a multiple of 2, 3, 4 and 5?

Notice that is a step is used by C it is also used by A so in determining the steps used exactly once we can ignore C. At this point you are left with the numbers 2, 3 and 5 which are all primes. Look at a step number between 1 and the bottom step and write this number in terms of its prime factors. How can you tell if it has been stepped on exactly once, more than once or never?

I hope this helps,

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