# Lights starting together.

A famous question from children's textbook on LCM (least common multiple) is: A bookstore named X Y Z has its named flashed on a neon light board. All the three letters flash for a duration of $$1$$ second before being put off. $$X$$ flashes after every $$2 \frac{1}{2}$$ sec, $$Y$$ flashes after every $$4 \frac{1}{4}$$ seconds and $$Z$$ flashes after every $$5 \frac{1}{8}$$ seconds. When will the full name of the bookstore be readable?

Now since this questions has been pigeonholed to the category of LCM in general textbooks a simple solution follows which gives $$\text{LCM}[5 \frac{1}{8}+1,2 \frac{1}{2}+1,4 \frac{1}{4}+1]$$ as the answer. I would be happy with the answer had the question been asking "when will all the letters light together" but the questions asks when they will all be lighting simultaneously. So, why are we not looking at situations in which one of them say $$X$$ starts lighting and in the $$1$$ second period of its being live, the other two start lighting. Is that too obvious that such a case won't exist[If it is please elaborate on how that can be seen] or is it just that since the question has been cloistered to one category, we don't pay much heed to it.

• Least common multiples are defined only for integers. – William Elliot Apr 2 at 22:55
• @WilliamElliot not true math.stackexchange.com/questions/44836/… – Roddy MacPhee Apr 3 at 0:44
• Is this question famous for being badly written? The question assumes a time interval, but there is no definition of when that interval begins - only when it ends. There is nothing that indicates where each letter is in its cycle when the interval begins. And the supposed answer appears to assume that the times listed are the intervals where the lights are off, but to me the question reads that they are the intervals between one lighting and the next. I.e., the 1 second on is part of those times. – Paul Sinclair Apr 3 at 2:35