GMAT Tip of the Week: Exponents the Denard Robinson Way

People talk about him the way they talk about Chuck Norris or the Most Interesting Man In The World:

When Denard Robinson scored a touchdown in the 2-minute drill the clock actually read 2:05*

Human reaction time can’t use a stopwatch fast enough to time Denard Robinson in the 40, so they have to time him in the 50. And his time is still zero.

Defenses have stopped trying to tackle Denard Robinson; they simply want to hug him.

Maybe it’s because Denard Robinson, quarterback of the Michigan Wolverines, is the most interesting man in the world. At least for now – in the past two weeks he’s vaulted to the top of nearly everyone’s Heisman Trophy rankings and topped the trends list on Google and Twitter. He broke several school records in his first game of the season, then broke them again in the second. With all due respect to the Dos Equis guy, sharks really should have a week dedicated to Denard Robinson. So, naturally, a guy like that should be able to teach you a thing or two about the GMAT, right?

(“If Denard Robinson took the GMAT, he’d score 801, but he doesn’t have to because any school lucky enough to admit him would waive that requirement and simply name their school the Robinson School of Business”)

It seems like every few years in college football there’s a player like Denard – Reggie Bush, Percy Harvin, etc. – who can play virtually any position on the field and is a constant threat to score each time he touches the ball. Maybe they line up at receiver, or go in motion. Maybe they line up in the backfield, or step up under center or take a direct snap. Wherever that player is, he’s dangerous – he’s versatile enough to hurt you running, catching, throwing, and the offense can hide him in different positions to keep you guessing.

For those players, the defense always has to look for where they line up as soon as the huddle breaks, calling out by jersey number “There’s 16″ and ensuring that everyone knows where he is.

The GMAT has such a “player” — a game changer that is as versatile and easy to hide as the greatest college football players of all time. That number is 0, and much like a safety or middle linebacker, you should always be looking for where 0 lines up on any given question.

0 is a complete game changer:

  • Multiply anything by 0 and you get 0.
  • Divide 0 by anything and you get 0.
  • Add or subtract 0 and nothing changes.
  • You cannot divide by 0 because it’s undefined (although the Chuck Norris/Most Interesting
  • Man quotes will lead you believe that some special person can divide by 0)
  • Take anything to the exponent of 0 (e.g. 6^0) and you get 1.
  • 0 is the only number that is neither positive nor negative; multiply it by -1 and it stays 0.

0 is easy to hide:

  • 0 is neither positive nor negative, so “x is positive” excludes 0 but “x is non-negative” includes 0.
  • 0 is an even number, but the only even without an opposite (e.g. 2 and -2)
  • 0 is a multiple of every integer (that integer * 0 = 0)
  • 0 literally means “nothing”, so it’s easy to forget about.

Because of all this, 0 carries all the traits of a dominant college football multi-threat, and you should never fail to consider 0 on any problem. Yesterday’s challenge question in this space relied heavily on properties of 0:

For integers x, y, and z, if (3^x)(4^y)(5^z) = 3,276,800,000 and x + y + z = 15, what is the value of xy/z?

3,276,800,000 is clearly even, so it could definitely be a multiple of 4. And it ends in 0, so it is definitely divisible by 5. But the sum of its digits do not equal a multiple of 3:

3 + 2 + 7 + 6 + 8 + 0… = 26

So the number is NOT divisible by 3. In problems like these with multiple prime bases and exponents, finding a base that is not represented in the overall number is a godsend. Because that number is not divisible by 3, the 3^x term cannot equal a multiple of 3. The only way for that to happen is for x to be 0, as that would make that term 3^0, which equals 1. That factors out that term, leaving 4^y * 5^z = 3,276,800,000.

Now, solving that problem could still prove difficult (although there’s a way to do it fairly easily if you look at it the right way…more on that in a second), but we don’t need to. Because we know that x = 0, and the question asks for xy/z, that means that we have a 0 in the numerator:

0*y/z =

And that makes the entire term 0. Therefore, the correct answer is 0.

Two footnotes:

1) Even with 4^y * 5^z = 3,276,800,000 there is a quick way to solve it. In order to end with exactly five 0s, this number needs to be divisible by exactly five 10s (10^5). You could even write it as: 32,768 * 10^5. The only way to have exactly five 10s is to have exactly five pairings of 2*5 (the prime factors of 10), so z must equal 5.

2) Denard’s finishing the two-minute drill with 2:05 left means he runs faster than the speed of light. Think about it.

Learn to love (and respect) the number 0, and just like a football coach you can game plan to defend against the versatile-and-dangerous game changer that the GMAT loves to feature. As for Big Ten defensive coordinators…maybe you should consult your MBA students for tips on defending Denard Robinson. (Number of times he’s tied his shoelaces? Zero.)

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