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Brian Galvin is the Director of Academic Programs at Veritas Prep, where he oversees all of the company’s GMAT preparation courses.
For anyone who wants to change the world, doing so can seem like an insurmountable task. The Catch-22, however, is that a world of individuals overwhelmed by the specific inability of each to change will miss out on opportunities to make incremental individual differences that sum to marked progress. Environmentalists like Al Gore (and his less-famous but probably more-impactful counterparts) have struggled with this concept for years – how can they mobilize individuals collectively toward an individually-overwhelming goal?
You’ll likely feel the same about some questions on the GMAT. Many questions feature uncomfortably high numbers for exactly that purpose – a freakishly large number can certainly overwhelm a test-taker as the task becomes much less tangible, the same way that problems that feature a mass of variables can do the same thing.
To combat these questions, think like an environmentalist: Think Globally, but Act Locally.
This mantra was enacted by the environmental community to encourage people to take pride in their small, individual activities that, if undertaken collectively, would extrapolate to mass change. On the GMAT, you’ll have opportunities to do the same, by testing the ways that numbers interact with small, “local” numbers, and then extrapolating those learnings to the larger, more abstract numbers in question.
Consider the question:
The sum of the digits of integer z is 186, and z = 10n – 4. What is the value of positive integer n?
A) 19
B) 20
C) 21
D) 22
E) 23
Taking 10 to any of the exponents provided in the answer choices will create a massive number hardly worth the time it would take to write on your noteboard (a process that will likely be fraught with error as you attempt to keep track of 19 zeroes). However, we know what 10n – 4 will look like simply by taking a few, smaller numbers that are easier to jot down:
10^³ = 1000, so 10^3 – 4 = 996
10^4= 10000, so 10^4 – 4 = 9996
As we look at these numbers, we can find some patterns. When n = 3, the result is 996, or a 6 in the units place preceded by two 9s (a total of 3 digits). When n = 4, we just add another 9 to the left, and we have 9996, or a 6 preceded by three 9s (a total of 4 digits).
We know that the larger the value of n the more zeroes we’ll add to that term, and then the more 9s we’ll have to begin z. Accordingly, z will be composed of several 9s with a 6 on the end. The question becomes just how many 9s we’ll need. Judging from the pattern above, we know that the value of z will have n number of digits – the same number of digits as the value of the exponent. We also know that one of those digits will be 6, while the others are 9.
To get the digits to sum to 186, we know that the units digit of 6 in 999….96 will account for 6, and the rest of the sum will be comprised of the 9s. if we take off that 6, the sum of the remaining digits – the 9s - is 180, and 180/9 means that we need 20 9s. To have 20 9s and then a 6, we need a total of 21 digits, and judging from the above pattern we know then that we need n to be 21. The correct answer is C.
On questions like this that deal with massive numbers, you’re well-served to find out what that large number will look like by using smaller versions of it to establish patterns and relationships that you can then carry forward. Much like an environmentalist, think about the big picture while taking smaller, more manageable steps, and you’ll find that the process is much smoother.
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