The post GRE Math – Exponent Challenge appeared first on Magoosh GRE Blog.

]]>What is the units digit of ?

Before you go scrambling for the google.com calculator, try the following: start small and look for a pattern.

First off, don’t let those question marks scare you. Remember, we are only looking for the units digit, so there is no need to know that , just that it ends in a 9. So when I go to figure out , all I have to do is multiply 9 by 3, which is 27, and more importantly, has a units digit of 7.

Now the key to these types of questions is not to multiply all the way up to desired exponent (which would probably eat up the remaining time in the section) but to find a pattern. Notice that the first four numbers, 3, 9, 7, and 1, repeat as soon as you get to . In other words, every exponent that is a multiple of 4 will end in a 1, just as ends in a 1. Every number that is one less than an exponent that is a multiple of 4, will end in a 7, just as ends in a 7.

Likewise, every even numbered exponent that is not a multiple of 4 will end in 9, just as ends in a 9. Therefore, will have a units digit of 9.

Now, let’s take the difficulty level up a notch. Indeed, that is mostly what the GRE quant section does: take familiar concepts and wrap them up in different—and often diabolical—ways.

The following question is very tough, but not tougher than something you might see test day. Take a stab at it!

In the equation, , n is a positive integer, and X and Y can be any integer between 1 and 9, meaning that X4 and Y7 are both two-digit integers. What are all the possible values of the units digit of p if p > 0?

[A] 1

[B] 2

[C] 3

[D] 4

[E] 5

[F] 6

[G] 7

[H] 8

[I] 9

The good news is we don’t have to worry at all about the value of X or Y. Remember the (?) above? The value of the tens digit is irrelevant to the units digit, e.g., , both end in 6.

Therefore, if the value of n is odd, then X4 will always end in 4 (the pattern is 4…notice how the units digits oscillates between 4 and 6). If n is odd, then the units digit of Y7 will be either 7 or 3. Taking the units digit of 4, from X4 when n is odd, we get 4-7 and 4-3, which will give me 7 and 1.

If n is even, then X4 will always end in 6, and Y7 will end in either 9 or 1 (6 – 9 = 7 and 6 – 1 = 5). Therefore, the possible units digits are 1, 5, and 7, or [A], [E], and [G].

The post GRE Math – Exponent Challenge appeared first on Magoosh GRE Blog.

]]>The post GRE Quant: The Difference of Squares appeared first on Magoosh GRE Blog.

]]>If you’ve been studying GRE for some time, you’ve very likely encountered the following:

.

You may, however, only seen the following equation in the context of algebra. Nevertheless, the formula above applies to number properties. Let’s take a look.

While you may be tempted to make a mad dash at it, calculating each of the squares, there is an easier way. Think of the ‘16’ as the ‘x’ and the ‘15’ as the ‘y’. Using the equation above we get:

.

Wow, that was much easier than figuring out the squares of both ‘16’ and ’15.’ Now, let’s try it for the second pair:

That leaves us with (29)(31). I know, you may be balking at my nifty little formula, thinking you still need to do some tedious multiplication. But despair not! We can still use the difference of squares formula:

is simply 3 x 3 add two zeroes: 900. Then we subtract the one and we get 899.

Next our two practice questions. The first question is not very different from the one above. The second one is more challenging and involves exponents.

1. ?

- 4
- 36
- 38
- 76
- 224

2. =

- 7
- 25
- 156
- 175
- 216

** **1.

76 – 72 = 4, Answer (A).

2.

.

The post GRE Quant: The Difference of Squares appeared first on Magoosh GRE Blog.

]]>The post GRE Exponents: Practice Question Set appeared first on Magoosh GRE Blog.

]]>Each of the math questions below is directly inspired by a question in the on-line Revised GRE test. I’ve provided an easier version of the question (#1) and a more difficult version of the question (#2).

My recommendation is to try the easier version first. Then, if you answer it correctly, click on the link, and take a stab at the actual Revised GRE question.

If you are able to answer that question correctly, then as prize – you get a fiendishly difficult question (#3). Okay, maybe that’s not a prize – but it is great practice for those aiming for the 90% on quant.

The good news is I have explanations. For the Revised GRE question, I have recorded an explanation video you can watch. Finally, it is a good idea to try the easy question before the medium one, and the medium question before the difficult one.

Good luck!

If , where n is a non-negative integer, what is the greatest value of ?

- ½
- 1
- 5
- 32
- 64

**Explanation:** Don’t think big – think small. That is the smaller n becomes the greater ½^n becomes. So what is the smallest value? You may be tempted to say 1, which would give us ½. But remember n = 0, because . Therefore **Answer: B.**

The “hidden zero,” as I like to call it, is a classic GRE math trick. So always keep your eyes open, especially when you see “non-negative integer,” which includes zero.

(5/4)^{-n} < 16^{-1}

What is the least integer value of n?

**Explanation:**

The best place to start here is by getting rid of the unseemly negative signs and translating the equation as follows:

(4/5)^{n} < 1/16

A good little trick to learn using 4/5 taken to some power is that (4/5)^{3} = 64/125, which is slightly — but only slightly — greater than ½. Therefore, we can translate (4/5)^{3} to ½.

(1/2)^{4} = 1/16

That would make (4/5)^{12} a tad larger than 1/16. To make it less than 1/16 we would multiply by the final 4/5, giving us n = 13.

The equation is true for how many unique integer values of n, where n is a prime number?

- 7
- 4
- 2
- 1
- None of the above

This problem can be difficult, indeed downright inscrutable, unless you take your time and process one piece of information at a time. Once you understand what the problem is saying, you should be able to solve the question relatively quickly.

**Explanation: **

The most important piece of info is n is a prime number. So do not start by plugging in zero or one. Neither is a prime. The lowest prime is 2. When we plug in ‘2’ we get:

This is clearly true. Thus we have one instance.

As soon as we plug in other prime numbers a pattern emerges.

is always a negative number if n is odd. Because all of the primes greater than 2 are odd, the number in the middle will always be negative:

Because in each case n is a positive number we can never have the middle of the dual inequality be positive, if n is an odd prime.

Thus the only instance in which the inequality holds true is if we plug in ‘2’, the answer is (D).

If you got that right – congratulate yourself. It’s a toughie.

For even more GRE questions, check out our GRE Quant problems with answers and explanations! And to find out where exponents sit in the “big picture” of GRE Quant, and what other Quant concepts you should study, check out our post entitled: What Kind of Math is on the GRE? Breakdown of Quant Concepts by Frequency.

The post GRE Exponents: Practice Question Set appeared first on Magoosh GRE Blog.

]]>The post GRE Math Basics: Quick Tips appeared first on Magoosh GRE Blog.

]]>**Prime numbers**

2 is the smallest prime number

2 is the only prime even number

1 is not a prime

3, 5, 7 are the only three consecutive odd integers that are each a prime.

**Digits**

In the integer 5,432, 5 is the thousands digit, 4 is the hundreds digit, 3 is the tens digit, and 2 is the units digit.

In a multiple-digit number, there are 10 possible units digits (0 – 9). In the first digit, i.e. the digit farthest to the left, there are only nine possible digits. A zero as the first digit would not be valid. For instance, 0,453 is not a four-digit number. It’s – if anything – the three-digit number 453.

**Divisibility Rules**

3: Add up the digits in any number. If they are multiple of 3, then the number is divisible by 3. For example, 258 is divisible by 3 because 2 + 5 + 8 is 15. And 15 is a multiple of 3.

4: If the tens and units digits of integer X is a multiple of 4, then integer X is divisible by 4. For example, 364 is divisible by 4, because 64 is a multiple of 4.

5: If a number ends in 5 or 0, it is divisible by 5.

6: If the rules to divisibility to 2 and 3 are fulfilled, then a number is divisible by 6. One hundred and eleven is not divisible by 6. Two hundred and forty six, because it is divisible by both 2 and 3, is divisible by 6.

9: If the sum of the digits of a number is divisible by 9, then that number is divisible by 9. For example, adding up the digits in 4,536, we get 4+5+3+6 = 18. Because 18 is divisible by 9, the number 4,536 is also divisible by 9.

**Odds (so to speak) and Ends**

Zero is an even integer

Integers can be negative numbers (-3, -6, -17, etc.)

0! = 1

x^0 = 1

The problems below can be solved by referencing the above. Good luck!

**Practice Problems**

1. What is the smallest integer that can be divided by the product of a prime number and 7 while yielding a prime number?

(A) 7

(B) 14

(C) 24

(D) 28

(E) 35

2. The sum of five consecutive even integers is 20. What is the product of the median of the series and the smallest integer in the series?

(A) 12

(B) 10

(C) 6

(D) 4

(E) 0

3. Which of the following integers is NOT divisible by 3?

(A) 624

(B) 711

(C) 819

(D) 901

(E) 969

Answers:

1. D

2. E

3. D

The post GRE Math Basics: Quick Tips appeared first on Magoosh GRE Blog.

]]>The post GRE Math Video Lessons: Exponents appeared first on Magoosh GRE Blog.

]]>Exponents are troublesome; fortunately, they do not need to be. What I plan to do in this series is show you the basics, so that you can move confidently through the intermediate level, and even the advanced level.

I’ve created three lessons, corresponding to basic, medium, and advanced. You can watch them here:

Once you have a sense of how the fundamentals of exponents work for each level, you can test yourself with a GRE problem. After all, knowing only the fundamentals does not mean you will answer a GRE question correctly.

Remember, the GRE is testing the way you think. That means the GRE will try to throw you off balance by presenting an exponents question that, at first glance, will give you pause. They expect you to be able to find an effective approach to the problem. Only then can you apply the fundamentals you’ve learned. In essence, the GRE is wrapping up a problem; unraveling a problem, so to speak, is half the battle.

**Level 1**

So now, I am going to take the fundamentals covered above, and I am going to wrap them up in a GRE problem. See if you can unravel the question below.

Practice Problem:

If , what is the value of y?

(A) 6

(B) 12

(C) 18

(D) 36

(E) 72

Answer: C

If you are even a bit unsure about any of the above, you will definitely want to watch the video.

**Level 2**

It’s going to be a little more difficult here, but if you are confident with level one, give Level 2 a try:

** **

Once again, I am going to take the fundamentals you’ve learned this far and I am going to turn them into a GRE problem:

** **

*, what is the value of n?*

*(A) **-2*

*(B) **0*

*(C) **1*

*(D) **2*

*(E) **4*

** **

Answer: D

If you were unable to answer the problem correctly, or if any of the above did not make sense, take a look at the video.

**Level 3**

Congratulations! You’ve already learned enough to answer most GRE exponent problems. In this level, you are going to learn more advanced techniques, perfect for those students looking to score above the top 85%.

Practice Problem:

If then n is a terminating decimal with how many zeroes after the decimal point before the first non-zero digit?

*(A) **2*

*(B) **3*

*(C) **5*

*(D) 7*

*(E) **9*

Answer (D). This question is about as difficult as any exponent problem you will see on the GRE. So, if you got it right, good job! If you are unsure about the solution to the problem, you can see the steps in the picture below (scroll down):

**:::::::::::::::::::::::::**

**:::::::::::::::::::::::::**

**:::::::::::::::::::::::::**

**:::::::::::::::::::::::::**

**:::::::::::::::::::::::::**

**:::::::::::::::::::::::::**

**:::::::::::::::::::::::::**

If you would like more practice on exponents, check out the Official Guide problems (there are a few exponent problems in the book). And, if you encounter any obstacles, don’t despair—I’ve recorded video explanations for every single problem in the Official Guide. For even more practice on exponents, and video explanations, give Magoosh a try.

The post GRE Math Video Lessons: Exponents appeared first on Magoosh GRE Blog.

]]>The post Math Basics: Exponents appeared first on Magoosh GRE Blog.

]]>One thing is certain on test day – you will see exponents. Another thing is also certain – those exponent problems will be meant to trick you. So, know your exponent properties, but also be aware that knowing the fundamentals alone won’t guarantee success. Tread carefully, and check your work. You should be able to answer most of these exponent questions correctly. Speaking of which, let’s see if you can answer the following three questions. They are not very difficult, only a little tricky.

1.

2.

3. If , what is the value of ?

For the first problem, do not simply add the exponents together. You can only do that when you are multiplying exponents, not adding them. Instead, notice how there are 4 of the same number, . Therefore, . Answer (C).

The next problem also asks you to add exponents, but first, you must multiply the exponents in parenthesis to the exponent outside the parenthesis, giving you: . Remember, that does not add up to . If you factor from both and , you get: . Answer D. Also, you can use approximation and eliminate those answers that are too small and too big. Answer choice E is far too big, and Answer choice A too small.

For the last problem, the best way to deal with it is by plugging in the answer choices. B, C, and D will all result in some funky irrational number because you are taking the square or cube root of the number. That leaves us with (A) -1 and (E) 1. Plugging in (A), you end up getting . Therefore, the answer has to be (E) 1. Remember, also, that any number taken to a negative exponent results in the reciprocal of that number, e.g. . And .

If you need more practice with exponents, look no further than Magoosh – both our products (oh, yeah, did I mention – we just released our new GRE product, over 600 practice questions waiting for you!) and on previous blog posts. Good luck!

The post Math Basics: Exponents appeared first on Magoosh GRE Blog.

]]>The post GRE Math Basics – Exponents appeared first on Magoosh GRE Blog.

]]>Exponents have long been the bane of many students. And that should come as no surprise—after all, there are negative exponents, fractions as exponents, and the terrible sounding exponents of exponents. Feeling comfortable with exponents in all their various (and nefarious) guises will definitely help you test day. But, for right now, let’s stick to the fundamentals of exponents.

Meet the Base

The base is the number underneath the exponent. In , two is the base and three is the exponent. means that you are multiplying (the base) three times: .

That’s the easy part. Now, what about when we are combining bases? Take a look:

The rule is as follows: if the bases are the same and you are multiplying them, add the exponents. So . That is I keep the base the same (I don’t multiply it) and then I add the exponents.

What about ? Well, the bases are different so we cannot combine them as we did before. In this case, you would just have to multiply the long way, .

There is one exception to this different base rule: if the exponents are the same but the base is different, you can multiply the bases. In , the bases are different but both are to the fifth power. In this case, we keep the exponent the same and we multiply the bases. Therefore, .

Let’s try a few more examples:

**Adding Bases**

What about when you are adding similar bases?

Can we add the and the ? The answer is an unequivocal no. The only way you can change the above number is by factoring: . This last step is relatively advanced, and I wouldn’t worry about it too much unless you are going for a high score. Just remember, if you are adding the bases—whether the same or different—then you cannot add the exponents.

Finally, what happens when you take an exponent to an exponent?

?

In this case, you multiply the exponents while keeping the base, 4, the same.

.

Therefore, not .

Think you got it? Okay see if you can answer the next three questions correctly!

** **

**Questions:**

**Answers:**

- Cannot mix bases; leave as is

The post GRE Math Basics – Exponents appeared first on Magoosh GRE Blog.

]]>The post Advanced Exponents–Problem Solving appeared first on Magoosh GRE Blog.

]]>The general approach is not too different—you want to look for a pattern. However this time you will have to be exact, instead of simply saying which side is bigger. One tactic the GRE employs is to make a problem look impossible to solve in 2 minutes. But don’t despair. All problems are solvable once you unlock them. So if your approach is taking too much time, step back and look at the problem from a new angle.

Try the following and see if you can solve it in fewer than 2 minutes.

What is the sum of the digits of integer x, where x = 4^10 x 5^13?

(A) 13

(B) 11

(C) 10

(D) 8

(E) 5

If you spotted the pattern early on, you may have only required a minute. But if you couldn’t unlock the problem, here is how you do so. Notice that 4^10 can be rewritten as 2^20. We can now express x as 2^20 x 5^13. The logic here is that 2 x 5 = 10. That is, 10 to any integer power greater than 1,will be a 1 followed by zeroes

So now let’s rewrite the problem again so we get 2^7 x 2^13 x 5^13. Combine 2^13 x 5^13 and we get 10^13. That is, we get 1 followed by 13 zeroes. If you are taking the sum, it’s straightforward: 1 plus 13 zeroes is 1. We are not done yet as we have the 2^7. When you multiply this out, you get 128. 128 x 1,000 thirteen zeroes is equal to 128 followed by the thirteen zeroes. Ignore the zeroes and we get 1 + 2 + 8, which equals 11. Answer B

The post Advanced Exponents–Problem Solving appeared first on Magoosh GRE Blog.

]]>