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If m is the product of all integers from 1 to 40, inclusive
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21 Jan 2019, 09:00
Let's talk strategy here. Many explanations on this forum focus blindly on the math. While that isn't necessarily bad, remember: the GMAT is a critical-thinking test. For those of you studying for the GMAT, you will want to internalize strategies that actually minimize the amount of math that needs to be done, making it easier to manage your time. The tactics I will show you here will be useful for numerous questions, not just this one. My solution is going to walk through not just what the answer is, but how to strategically think about it. Ready? Here is the full "GMAT Jujitsu" for this question:
First of all, it is worth highlighting one of the commonly-found traps embedded in this question. I call it "Mathugliness" in my classes. (Get it? It's math. It's ugly. It acts like a thug. But, like most thugs, it's main game is to intimidate.) This question deliberately uses obnoxiously-large values, in the hopes of scaring novice test takers. After all, if you were to actually work out the "product of all integers from 1 to 40, inclusive", this would result in a massive number called \(40!\) (or "40 factorial.") It would turn out to be:
\(40*39*38*37*36*35*34*33*32*31*30*29*28*27*26*25*24*23*22*21*20*19*19*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1=\)
\(815,915,283,247,897,734,345,611,269,596,115,894,272,000,000,000\)
Obviously, there is no way you are going to do this math. When you see "Mathugliness" on the GMAT, don't panic. There will always be a quicker, more conceptual way to look at the question, allowing you to avoid most of the messy math. The key is to focus on exactly what the question is asking, looking for logical leverage to strategically attack the question. In this case, the problem asks for the "greatest integer \(p\) for which \(10^p\) is a factor of \(m\)." In other words, "how many times could we factor a \(10\) out of \(40\) factorial?"
Of course, a possible trap would be to just think of the multiples of \(10\)s in the factorial:
\((40)\) \(*39*38*37*36*35*34*33*32*31*\) \((30)\) \(*29*28*27*26*25*24*23*22*21*\) \((20)\) \(*19*18*17*16*15*14*13*12*11*\) \((10)\) \(*9*8*7*6*5*4*3*2*1\)
There are four multiples of \(10\) in \(40!\), but this isn't even one of the answer choices. (To be honest, it should be. This is a missed opportunity on the part of the Test-maker!) But if we think about a "\(10\)" as a product of its parts (or \(2*5\)), it is obvious to see that there are a lot more \(10\)s embedded in the product. So the real question we should ask is: "how many \(2\)s and \(5\)s can we find in \(40!\)?"
There are A LOT of \(2\)s. After all, every 2nd integer will be divisible by \(2\). Since we are looking for the "greatest integer \(p\) for which \(10^p\) is a factor of \(m\)," the number of \(2\)s will not be our limiting factor. (In fact, we could actually factor out a "\(2\)" thirty-six times from \(40!\), but there is no way we could match up all of those \(2\)s with a corresponding \(5\) to create \(10\)s.) There are a lot fewer \(5\)s embedded in the expression.
Of course, we could count the multiples of \(5\), but doing this would also be a trap, based on a poor assumption:
\((40)\) \(*39*38*37*36*\) \((35)\) \(*34*33*32*31*\) \((30)\) \(*29*28*27*26*\) \((25)\) \(*24*23*22*21*\) \((20)\) \(*19*18*17*16*\) \((15)\) \(*14*13*12*11*\) \((10)\) \(*9*8*7*6*\) \((5)\) \(*4*3*2*1\)
If we were to merely do this, our answer would be \(8\). This is the most commonly chosen wrong answer. The way around this bad logic is to realize that the number of "multiples of \(5\)" embedded in a factorial isn't exactly the same thing as the number of \(5\)s. \(25 = 5*5\). Thus, it can contribute two \(5\)s to the total amount. The number of \(5\)s we could factor out of \(40!\) is therefore \(9\), and the answer is C.
Now, let’s look back at this problem through the lens of strategy. Your job as you study for the GMAT isn't to memorize the solutions to specific questions; it is to internalize strategic patterns that allow you to solve large numbers of questions. This problem can teach us patterns seen throughout the GMAT. The primary pattern this problem demonstrates is "Mathugliness" -- whereby the GMAT tries to "flex on you," using obnoxious or repetitive math to make the problem seem harder than it actually is. When you encounter Mathugliness, get excited. Look for leverage in the problem that will allow you to solve the problem conceptually, instead of working out all the math. In this particular problem, solid leverage can be found in the phrase "greatest integer." Questions that ask for maximum values are prime candidates for critical-thinking. You just need to look at how the problem is structured and determine what would limit or define the value you are looking for. Determine what those limits are, and you have your answer. And that is thinking like the GMAT.