How to add lots of integers quickly on the GMAT

When a question asks you to add a large number of integers, it can seem as if the answer may take some time to find. But the GMAT is all about smart shortcuts. Here's a quick method that should make you happy to see such questions on test day.

Consider the following question from the

Economist GMAT Tutor:

For every positive integer n, the nth term of a sequence is the sum of three consecutive integers starting at n. What is the sum of terms 1 through 99 of this series?Rules for finding the sum of this sequence of integers quicklyRule #1: The sum of a sequence of integers is the average of the sequence of integers multiplied by the number of terms.If this rule sounds familiar, it is because it is simply another version of the formula for finding an average:

\(Average = \frac{Sum \ of \ terms}{Number \ of \ terms}\)

Now, how do we determine the average of a sequence of integers?

Rule #2: The average of a sequence of integers is the average of the first and last termsApplying the rules to find the sum of the sequenceHow do we apply these useful rules to this question?

First, calculate the average of the first and last terms.* The first term is the sum of 1, 2 and 3 = 6

* The last term is the sum of 99, 100 and 101 = 300

* The average of the first and last terms = (6 + 300) / 2 = 306 / 2 = 153

Second, multiply the average by the number of terms.* There are 99 terms

* Therefore, the answer to our question is 153 x 99

Note that a quick way to calculate this without a calculator is to multiply 153 x 100 and subtract 153.

* 153 x 100 = 15,300

* 15,300 – 153 = 15,147

And that is our final answer!

You can see how the application of simple rules makes questions that seem difficult much easier. Your task is to assemble as many such rules as you can.

How to handle square roots in GMAT Properties of Integers questions

Questions involving properties of integers in combination with square roots may seem difficult at first glance, but with the application of the appropriate rule, they can be solved quickly.

Consider the following question from

Economist GMAT Tutor:

What is the smallest positive integer x, such that √(392x) is an integer?

A. 2

B. 4

C. 7

D. 8

E. 14This question is discussed

HEREThe rule you must understand to solve this square root questionThe rule may sound difficult to understand, but here it is:

The prime factors of any integer that is a power of another integer come in pairs, triplets, quadruplets etc. according to the power. If a is an integer, and a^2 is an integer, then the prime factors of a^2 must come in pairs.

Let’s think about that for a minute. Consider another example. If instead of considering a^2, we consider a^3. Imagine we are told that a and a^3 are integers. What could we conclude from this? For this to be true, the prime factors of a^3 must come in threes. For example, if a = 2, a^3 = 8. Thus, a and a^3 are integers. The prime factorization of 8 = 2 x 2 x 2 – a group of 3.

Applying this rule to the current problem* Firstly, find the prime factors of 392.

* These are 2 x 2 x 2 x 7 x 7.

* The 7s are a pair, but the 2s are not.

* If you multiply 392 by 2, you get 784.

* The prime factorization of 784 is 2 x 2 x 2 x 2 x 7 x 7.

All the prime factors are now in pairs, and all we needed was an extra 2. Thus, the square root of 784 will be an integer. Our answer is therefore A.

Remembering this rule about the powers of integers will help you to solve seemingly very difficult questions quickly.

How to add multiples on the GMAT

Sometimes questions involving properties of integers and calling on you to add multiples can seem complex. However, if you apply the right rule, you can solve such questions quickly.

Consider the following question from the

Economist GMAT Tutor:

Zeta took several pictures with her new digital camera. Each picture was saved as a file, the size of which depended on the picture's resolution. A low-resolution image requires a 0.5-megabyte file, and a high-resolution image requires a 0.9-megabyte file. If the total size of all the files was 8.1 megabytes, which of the following is a possible number of pictures Zeta took with her camera?

A. 11

B. 12

C.13

D. 14

E. 15This question is discussed

HERE.

How to add multiples quickly* Rewrite the question in the form of an equation:

* 0.5x + 0.9y = 8.1

* Multiply by 10 to remove the decimal points

* 5x + 9y = 81

With the equation written like this, we are solving for x + y, or the total number of pictures.

Apply the appropriate rule:

A multiple of x + a multiple of x = a multiple of x.81 is a multiple of 9. 9y is also a multiple of 9. Therefore, 5x must also be a multiple of 9.

Since 5x must also be a multiple of 9, x must be a multiple of 9. The only possible value for x can be 9, since 18 or any larger multiple is larger than any of the answer choices.

So, if x = 9, let’s solve for y:

* 5(9) + 9y = 81

* 45 + 9y = 81

* 9y = 36

* y = 4

* With x = 9 and y = 4, then x + y = 13, which is answer choice C.

In summary, applying the rule that

when you add two multiples of x, the result will be a multiple of x helped us to solve this question quickly. It’s a good rule to remember!

How to solve GMAT sequence questions

An important area of GMAT quant is sequences. Some little-known information about these arranged sets of numbers will save you a lot of time in answering questions in this area.

Consider the following question from Economist GMAT Tutor:If M is the set of all consecutive multiples of 9 between 100 and 500, what is the median of M?How to solve this sequence question1. Establish the first and last numbers in the set.

2. You should know in your head that 12 x 9 = 108. If you don’t, this is an opportunity to realize that knowing your times tables in your head is one of the most useful time-saving methods on the GMAT. 108 is therefore the first number in our set, as it is a multiple of 9 and greater than 100.

3. Divide 500 by 9 and you get a little more than 55.

4. Multiply 55 x 9. You get 495. The last number in our set is therefore 495.

An important rule for sequencesNow here comes the the trick. The important rule for you to remember is that the median of a sequence of numbers is the same as the average. Secondly, the average in a sequence of numbers can be obtained by adding the first and last terms of the sequence and dividing by 2.

Therefore:1. 108 + 495 = 603

2. 603 / 2 = 301.5

3. Therefore, 301.5 is the median of Set M.

Plugging in numbers to solve sequence questionsDon’t forget the strategy of plugging in numbers when solving sequence questions. This strategy can be used in most areas of GMAT quant, and it applies to sequences, as well.

Consider the following question from the Economist GMAT Tutor:The total number of plums that grow during each year on a certain plum tree is equal to the number of plums that grew during the previous year, less the age of the tree in years (rounded down to the nearest integer). During its 3rd year, the plum tree grew 50 plums. If this trend continues, how many plums will it grow during its 6th year?Plug in the numbers:1. In its fourth year the tree grew 50 - 3 = 47 plums.

2. (Notice that this is not 50 - 4, because the age of the tree is still 3 years, not yet 4 years.)

3. Continue this method

4. 5th year: 47 - 4 = 43

5. 6th year 43 – 5 = 38

6. The answer is therefore 38.

Remember these simple rules and strategies about sequences and you’ll have more time for the next question on the quant section!

How to solve GMAT quant questions using the median in a sequence

Here we’re sharing the most efficient way to answer GMAT Quant questions that involve calculations based on the median of a set of values.

Consider the following question from an

Economist GMAT Tutor lesson:

If in a certain sequence of consecutive multiples of 50, the median is 625, and the greatest term is 950, how many terms that are smaller than 625 are there in the sequence?

A) 6

B) 7

C) 8

D) 12

E) 13This question is discussed

HERE.

How to use the median to solve this questionRemember that by definition, the median is equidistant from the first and last values in a sequence.

* Calculate the difference between the last value and median

* 950 – 625 = 325

* Subtract 325 from the median to get the first value in the sequence

* 625 – 325 = 300

The multiples of 50 that are smaller than 625 are therefore 300, 350, 400, 450, 500, 550 and 600 – a total of 7.

Hence, B is the answer.How to spot the trap in these questionsIf you were not paying attention to detail, you may have thought that 625 was a value in the sequence. You could quickly have calculated that there are 13 multiples of 50 between 300 and 950 (a difference of 650). You may have then thought that there would be six multiples of 50 below the median and 6 above the median. In that case you would have answered A.

Note that while 625 is the median of the sequence, it is not actually a value in the sequence. It is the average of the 6th and 7th values in the sequence – the average of 600 and 650. You would recognize this quickly because 625 is not a multiple of 50.

Be particularly careful on test day if you have a tendency to quickly land on answer choices like A. The test makers are often trying to get you to answer questions too quickly. When they are doing that, they will often put the incorrect answer choice right in front of your eyes – Answer A.

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