How many multiples of 3 are there between 15 and 81, inclusive?
For a problem like this, it may be tempting to simply count out all the multiples of 3 by hand. While this would work, it would certainly not be the most efficient method. We can use a pattern to simplify our work. The first question we should ask ourselves is, “How often do multiples of 3 show up?” The answer to this, unsurprisingly, is that multiples of 3 show up every third integer. This tells us that we will want every third integer between 15 and 81. To calculate this, we find one-third of our total amount of integers:
Unfortunately, we are not quite finished. When we subtracted 15 from 81 above, we inadvertently left out one of our integers, specifically one of the two endpoints. Since 15 and 81 are both multiples of 3, we can’t leave either one of them out. As such, we have to add that missing integer back into our total:
This gives us our final answer of 23.
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