**It’s not about what you DON’T know? It’s about what you know… Part I **

Ever look at a problem and the first thing that comes to mind is, “I don’t know?” Well if you are anything like I used to be, then that phrase “I don’t know” popped in my mind a lot. A LOT. Like when I was asked by my friends back in the U.S. “what the hell are you going to do in Argentina?” Or my family members asking, “when are you going to get married?”

I’m joking, of course, I don’t know the answers to those questions (except for the Argentina one – answer: enjoying the best parts of my life right now in Buenos Aires). But, what about a question like “how can I get a promotion at work” or “how can I make my team more efficient?” or even “how am I going to complete this project on time?”

Now when someone responds to a question with “I don’t know,” it can lead them down a very pessimistic route. Let’s examine someone’s thought process here for a second. Let’s say this person came across a particular tough GMAT problem, like say this one that I found on GMATClub. And for the story’s sake, let’s call this person: Yaniv.

**700 – 800**

*Set S consists of numbers 2, 3, 6, 48, and 164. Number K is computed by multiplying one random number from set S by one of the first 10 non-negative integers, also selected at random. If Z=6 ^{K}, what is the probability that 678,463 is not a multiple of Z? *

*A: 10%**B: 25%**C: 50%**D: 90%**E: 100%*

So Yaniv’s first thought is of course, “I don’t know.” (My first thought was, “WTF is this?”) Now, I want to be perfectly clear here.

**This is normal. **

Think about it, when’s the last time you did math in the last five years that was more complex than knowing which formula to use when using Excel to calculate something. Well, if you are like most of my students and Yaniv here, your first thought should most definitely be “I don’t know.”

But that’s a dangerous thought and here is why.

What’s Yaniv’s next thought going to be? Well, “I don’t know,” leads to a series of negative thoughts, such as “how can I solve this problem in two minutes” or “I have no clue, where to begin?” Next, poor Yaniv is thinking the most dangerous of thoughts, “there is no way I can do this.” Once Yaniv has resided himself to the notion that he can’t solve this problem, he has no choice but to haphazardly waste time trying to figure out a solution or do the second wisest thing – guess and move on.

Now let’s change the narrative here for a second. Let’s say Yaniv doesn’t say “I don’t know.” Let’s pretend that he instead asks himself, “wait, what do I know here.” Now that’s an interesting concept. The question, “what do I know,” leads us to a more positive thought progression here. Because what you know could be anything, even something very small. Because understanding even the smallest concept of a difficult problem is taking a small step in the right direction. And like every mountain Yaniv climbs in his life, it begins with a small step.

Now, Yaniv began to wonder what he knows about the problem he just read. This time he just focused on the first line of the problem:

*Set S consists of numbers 2, 3, 6, 48, and 164.*

So the first thing he could figure out was the phrase, “consists of numbers.” He knew that meant that there was a group of numbers or something that had numbers in it. So a group of numbers is what they mean when they say “set of numbers.”

So he then wrote down:

S= {2, 3, 6, 48, and 164}….

“Ok now what to do,” he thought. He decided to see what else he knew or what he could figure out. He read the next phrase.

*Number K is computed by multiplying one random number from set S by one of the first 10 non-negative integers*

“Number K? What does that mean…well K is like some variable here, but wait what’s a variable?”

Thinking back to days when he had no hair on his chest, Yaniv remembered from his old algebra class that a variable is just any letter or a symbol that represents a number. So he decided to reread the question and replace K with an actual number like “2.”

So now it appeared to say:

*The number 2 is computed by multiplying one random number from set S by one of the first 10 non-negative integers*

“Ok, so 2 is computed by, which is like saying, 2 equals one number from this set S, by some other number.” He then thought about what “by” meant and gave himself an example of when he heard “by” used in some math sense.

Since Yaniv loves doing carpentry work for free, (need help fixing stuff around the house? Feel free to email Yaniv anytime. He loves giving up his Saturdays to do that shit) he thought about square meters like 20 x 20. For him, that just meant to multiply 20 with 20 to give him 400 square meters. And just like that Yaniv had figured out that “by” meant multiplying by relating it to something he already knew about.

So now he’s figured out that 2 equals one number from this set S, multiplied by one of the first 10 non-negative integers.

“WTF is an integer?” he thought.

Hahaha, no I’m kidding, of course, Yaniv knew that an integer is a whole number and not a decimal or a fraction. I mean who doesn’t know that??

**Note: even if you don’t know what a variable or an integer is, don’t let that deter you from doing something challenging like preparing for the GMAT. It will take you ten seconds to google the shit out of it. Seriously.
**

So then he looks at the phrase “non-negative integer” and asks himself:

“Why did they say non-negative? Wait, what’s a negative number?” he then thought.

“That’s easy, -1, -2, -3 and so on,” he remembered.

“Ok, but why not say positive and not non-negative?” And then came his ah ha! moment. “Because those tricky bastards don’t want me to include 0.”

0 isn’t negative, it ain’t positive but it sure as hell ain’t negative (it’s neutral). He then realized that the first ten non-negative numbers must be:

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9} NOT {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Ok so now he’s written down:

K (*because he realized that K is just some number like 2*) = one number from S= {2, 3, 6, 48, and 164} multiplied by one other number from some set that has {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and let’s call that set T (why T? It doesn’t matter, Yaniv picked it only because it came after letter S. Who cares what it’s called!).

Now finally at the last part of the problem, Yaniv is in the zone, he’s a problem-solving machine and he can’t be stopped. So he reads the last sentence which says:

*If Z=6 ^{K}, what is the probability that 678,463 is not a multiple of Z? *

And now the unstoppable problem-solving machine has hit a road bump. This sentence gives his simple mind multiple headaches.

Words like “probability, multiple, and of course “ Z=6^{K}.” So what does Yaniv know here? He starts to think about the word “multiple.” So now he begins to ponder on what multiple means by concentrating on a small number like 4. He realizes that 4 is simply 2 x 2 and comes to the conclusion that 2 is a multiple of 4. Meaning if one number, say “y,” can be divided by another number, say “x,” and the result is an integer, then x is multiple of y. Or in the example above 4 divided by 2 is 2, so 2 is a multiple of 4. Ok, one down. Two more to figure out.

He decides to tackle probability next. What does he know about that word? He knows that the weatherman said there was an 80% chance that it will be sunny this weekend and that meant there is a good chance that the awesome Frameless Event Party by Dan Thompson and Nico Reynolds isn’t going to be rained out.

Ok, so 80% is like saying there is an 80/100 chance of no rain and all fun to be had. So he realizes that if there is an 80% of it being sunny then the chance of it raining has to be 20%, then 80% and 20% must equal 100%. Well, Yaniv just stumbled upon the famous probability formula which is:

“Desired Outcomes” divided by “Total Possible Outcomes.”

Where in the case of sunshine, the desired outcome is 80 and the total possible outcomes of rain or shine is 100.

Now on to the hard part: “Z=6^{K}.” Before he starts freaking out, he remembers that he can replace “K” with a number to help him understand. So he says that K = 3, and then reads Z=6^{K} as Z=6^{3}. Let’s see, he ponders, I know that x^{2} is x time x or 2^{2} is 2 x 2. So then 6^{3} = 6 x 6 x 6. Ok great, he thinks. Whatever number K is, then there are K number of 6s multiplied together.

Now he checks the last part of the sentence again and it says this:

*the probability that 678,463 is not a multiple of Z? *

So Yaniv knows that Z really is just some number made up with a bunch of 6s. So instead of getting frightened by the ridiculous number, 678,463, he simply asks: Is this number even a multiple of 6? Or does this number have any 6s in it?

Nope. How does he know? By simply doing the long division and actually dividing 6 into 678,463 and seeing that it does not divide cleanly. (If Yaniv was clever, but he is not, he would have simply realized that 6 =2 x 3, and 678,463 has to be divisible by 2 and 3. It’s not an even number, so it’s not divisible by 2, and if you add the digits of 678,463, you get 6 + 7 + 8 + 4 + 6 + 3, which equals 34. 34 is not divisible by 3.)

So he then goes back to what he learned about probability and realizes that there is no way 6 to the power (6^{K}) will ever be a multiple of 6. So that must mean there is 0% probability that 678, 643 is a multiple of whatever number Z comes out to be. Or to answer the question, there is 100% probability 678, 643 is a multiple of Z.

Boom Baby! He got it!

E. 100% He’s kicking himself now because he probably would have guessed that answer. But before Yaniv clicks on answer choice E, he asks himself, was that too easy? Well, was it? Cause this problem sure as hell doesn’t look easy. “Let me read it again”, he says to himself.

* Set S consists of numbers 2, 3, 6, 48, and 164. Number K is computed by multiplying one random number from set S by one of the first 10 non-negative integers, also selected at random. If Z=6 ^{K}, what is the probability that 678,463 is not a multiple of Z? *

*A: 10%**B: 25%**C: 50%**D: 90%**E: 100%*

He keeps coming back to those set of words “*10 non-negative integers.*” Why did the author write that? Why didn’t he simply write the first 10 positive integers?

Why? He thought…then he got it. It had to do with the 0. The 0 was clearly out of place here. So he wondered about 6 and wondered what 6^{0}. Now this is where it gets a little tricky. You have to know that 6^{0} is 1 and not 0. (Why? JUST BECAUSE! THAT’S WHY! For all you math geeks like me, you can see the proof here.)

Let’s assume divine inspiration struck Yaniv, and he remembered that 1^{0}=1 from his old math classes. So now, Yaniv puts it all together. 678,463 is divisible by 1! All numbers are divisible by 1. So if K=^{0} and 6^{0} = 1, then we have a possibility that number Z is divisible by 6^{K}. 6^{K} had to equal 1.

So Yaniv went back to sets S and T that he had previously scribbled down. One random number from Set S:

S= {2, 3, 6, 48, and 164}

Had to multiply with one number from Set T:

T= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

to give us 0. So he then noticed the only way he could possibly get a 0 was if the number he picked from set T was 0. So he asked himself, what’s the probability of selecting 0 from Set T. Well he thought, there is only one chance out of ten numbers that he would select a 0. So I have a 1 in 10 chance or a 10% percent probability. Since it didn’t matter what number he selected from Set S since any number multiplied by 0 is of course 0, he finally came to the conclusion that the probability that Z was a multiple of 643,478 is 10%!

Eureka! He exclaimed, then proceeded to rip off his clothes and run through the streets of Buenos Aires naked like Archimedes. After an eventful night (not like any other weekend here, I might add), Yaniv returned the next morning to find out that he actually got the problem wrong. You see, Yaniv made a crucial mistake. He forgot to re-read the last sentence of the question one more time. It said:

* If Z=6 ^{K}, what is the probability that 678,463 is not a multiple of Z?*

You see it? Yup, NOT A MULTIPLE OF Z. The GMAT trolls got him again. If 10% was in fact the probability that Z (or 1) was a multiple of 678,643, the opposite would be that there is a 90% chance it is not. (The answer is D. 90%)

The moral of this long story, is simply to change your mindset. Try it out for a week. Whenever you have a problem that you have no idea how to solve, try asking yourself “what do I know” instead of “I don’t know.” You’ll be surprised what you can accomplish when you change the questions you ask yourself. It all starts with what you know.

## Comments 1

I read all this in a friend’s voice in my head while a long “subte” ride and laughed at the little remarks on poor Yaniv’s mathematical time-travel adventure…im deffenetly taking more time next time “I unknowingly decide to not know something”