How Fast Can You Solve this GMAT Prep Quant Problem?
Try this problem from the free problem sets that come with the GMATPrep software and see whether you can spot the most efficient solution.
Here is the problem:
* If a is a positive integer, and if the units digit of [pmath]a^2[/pmath]is 9 and the units digit of [pmath](a+1)^2[/pmath]is 4, what is the units digit of [pmath](a+2)^2[/pmath]?(A) 1
(B) 3
(C) 5
(D) 6
(E) 14
What did you think? If you thought it was okay (or even easy), then dont keep reading yet. First, go back and see whether you can find any more efficient solution paths. Then, once youve decided on the best solution, see whether you can articulate out loud not just what to do but how you knew to take each step along the way. If you can articulate that well enough to teach someone else, then you have mastered this type of problem.
If you thought this was hard, read on.
Step 1: Glance Read Jot
Take a quick glance; what have you got? PS. Some given information about a. A question.
Jot the given info on your scrap paper:
a = + int
[pmath]a^2[/pmath] units dig 9
[pmath](a+1)^2[/pmath] units dig 4
[skip a few lines here]
[pmath](a+2)^2[/pmath] units dig = _________?
I skip a few lines between the givens and the question so that I can do the work in between and then run into my reminder of the question. That way, Im less likely to get lost or solve for the wrong thing.
Step 2: Reflect Organize
They keep talking about a, but they dont actually give any real values for it. What to do?
Is there only one possible value for a, or are there multiple? How do you know?
The question stem never provides a real value for a, nor do the answer choices. Everything hinges around the units digit. As a result, any value of a that follows the given constraints should yield the same answer; this is a variation on the classic Smart Numbers set up. You can pick your own real number for a; as long as you follow the given constraints (and do the math correctly), you should arrive at the right units digit.
Step 3: Work
So what number should you pick? Obviously, an integer.
Then, lets seethat integer squared needs to have a units digit of 9. What integers square to a units digit of 9?
[pmath]3^2[/pmath]= 9
Does the next constraint work?
Okay, when a = 7, then [pmath]a^2[/pmath]has a units dig 9 and [pmath](a+1)^2[/pmath]has a units dig 4. What is the units digit of [pmath](a+2)^2[/pmath]?
[pmath]9^2[/pmath]= 81
The units digit is 1.
Okay, so what next? Is the answer 1? Or do you need to try another number?
In this case, it turns out that you dont need to try another number; the answer does always have to be 1. Why?
Lets say that you did try another number and, this time, the answer was 3. Then what would the correct answer be? You cant pick both (A) and (B); the problem has to have one distinct correct answer. If two answers worked, the problem would be brokenand the GMAT isnt going to give you a broken problem.
The correct answer is (A).
Key Takeaways: Smart Numbers
(1) Its very important to recognize when you can make a problem easier by picking real numbers to solve. The problem above can be done in a theoretical way, but it would be pretty painful for anyone but a mathematician (or someone who is much better at math than we need to be for the GMAT).
(2) You can try your own real number(s) whenever the problem talks about some number but never provides a real value for that number anywhere in the problem or in the answer choices. That can even work when the problem gives you part of the value (the units digit) but not the full value.
(3) On these theory problem solving problems that can be solved using real numbers, you only have to try one number, since there must be one discrete answer. Do make sure that you follow all of the given constraints; some of the wrong answers will be built off of messing up those constraints.
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.