Now that the World Cup is over and the majority of the world’s population has forgotten again about the Southern Hemisphere, it’s safe to say that most of us are enjoying the middle of summer. Summer is more than just a season between the solstice in June and the equinox in September; it’s a state of mind and a way of life. Close your eyes and just think of “summer” — just the notion of it implies to most of us a sense of happiness, relaxation, and comfort. Take Christmas carols out of the mix and summer is easily the most musically-written-about season of all. Admit it — you have DJ Jazzy Jeff & The Fresh Prince’s “Summertime” in your head right now!
Summer, and the music that it inspires, can teach us about how to succeed on the GMAT and in other pursuits. Peak performance tends to come when stress levels are lower, when we feel calm and confident, and when we’re enjoying a positive frame of mind. Since summer is generally associated with all of those feelings, embracing our inner summertime can be instrumental in achieving peak performance. As Nat King Cole wrote it, we should “roll out those lazy, hazy, crazy days of summer…” Here’s how:
Focus on the lazy!
The GMAT, like life itself, will try to encourage you to work too hard. Also like life itself, however, the GMAT will reward you for finding a simpler, easier way. The key to that is telling yourself that you don’t want to work too hard, and that you can find a way to be lazy (which, if you’re smart about it, is just another word for “efficient”).
Consider this tough Data Sufficiency problem:
What is the remainder when integer n is divided by 10?
1) The tens digit of 11^n is 4
This statement is a tough one — at first glance it looks impossible, as we’re pretty good with units digit properties but don’t have very many (if any) hard-and-fast rules for digits to the left of that. It almost seems as though you have two choices: either assume that “n could be anything so there’s no way that this works,” or start multiplying out 11 after 11 to see if you find a pattern. The multiplication sounds just awful — after 11^1 as 11 and 11^2 as 121, those values get big in a hurry, and multiplication can be incredibly time consuming.
But does it have to be? After all, this is the GMAT, a carefully-written test that rewards efficient (or just plain lazy) ingenuity. You can’t simply assume that “statement 1 doesn’t work because n could be anything” — the GMAT’s authors specifically wrote that statement for a reason, and they’re not apt to give you a “throwaway.” But you also don’t have to mindlessly grind out calculations, either — the authors want to give you that option, but they also write these carefully enough to give you an out if you’re clever (or lazy) enough to look for it. Here’s that out:
Multiplying by 11 is the same as multiplying by 10 and adding that to the original number (11 = 10+1).
Watch how the exponents of 11 increase:
11^1 = 11
11^2 = 11*10 + 11 * 1 = 110 + 11 = 121
11^3 = 121*10 + 121*1 = 1210 + 121 = 1331
11^4 = 1331*10 + 1331 = 13310 + 1331 = 14641
11^5 = 14641*10 + 14641 = 146410 + 14641 = 161051
and so on…
Notice any patterns?
First, hopefully you’ve found that multiplying by 11s was easier than expected -multiplying by 10 just means adding a 0 on the end of the initial number, and then adding that original number back in to account for the 11th value.
The units digit is always 1, which should stand out but also be expected (when multiplying two numbers that end in 1 you’ll always get a number that ends in 1).
The tens digit is also interesting though — look at how it increases: 11, 121, 1331, 14641, 161051. The tens digits go from 1, 2, 3, 4, 5…. in lockstep with the value of n, the exponent.
Here is where the GMAT provides you with another decision point. You can either assume that this pattern will continue (not a terrible assumption since it’s worked for five values of n in a row, albeit all single-digit values of n which may make them too similar to be proof of an infinite pattern) or try to prove it to yourself. Proving it may seem to take too much math — we can continue the process we’ve been doing to make the multiplication easier, but the bigger the numbers get the more time-consuming and error-prone that process will be. Is there an easier way?
Go back to the process we’ve used for multiplication by 11:
Multiply by 10 — which means just add a 0 on the end — and then add the original number back.
Well, every “original” number we have to multiply by 11 has a units digit of 1, which means that when we add the zero on the end of that number it will end in 10, and have a units digit of 10. When we add that back to the original number, then, we’re simply increasing the tens digit by adding 1. This pattern will hold infinitely — we’re going to add 1 each time to the tens digit, which means that whenever n ends in a 4, then the tens digit will be a 4. Therefore, the statement is sufficient.
Note that you don’t need to (and probably shouldn’t) memorize this rule! There’s a process for identifying unique number properties when they arise:
Three-Step Process for Unique Number Properties
1) Identify that you are being tested on Number Properties: When a question deals with incredibly large values (or potential values), you’re likely to be able to use a number property; when a number asks about units digits, tens digits, etc., you’re almost certainly being tested on number properties.
2) Look for a pattern: When you’ve identified that you’re likely to need to use a number property, try to find a pattern using smaller numbers that you can extrapolate to larger ones.
3) Determine why: Once you’ve identified a pattern, time permitting, try to reason why the pattern holds. If you don’t have time, but you’ve established a recurring pattern, you may need to simply assume that it’s much more likely than not that it will continue. But if you can afford the time, and definitely during your homework when time isn’t a factor, try to rationalize the reason for the pattern.
The more that you embrace this method, the easier it becomes, and the lazier (er, more efficient) you can be. Finding patterns, and in doing so finding ways to accomplish difficult tasks quickly, is a highly-rewarded thought process on the GMAT, as it is in life. Embrace that summertime spirit of relaxation and calm, and enjoy the rewards that laziness can bring.
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