The sum of 4 different odd integers is 64. What is the value of the

The sum of 4 different odd integers is 64. What is the value of the greatest these integers?

(1) The integers are consecutive odd numbers --> x + (x + 2) + (x + 4) + (x + 6) = 64. We can find x. Sufficient.

(2) Of these integers, the greatest is 6 more than the least --> least = x and greatest = x + 6. Between x and x + 6, there are only 2 odd integers x + 2 and x + 4, so we have the same case as above. Sufficient.

Answer: D.
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We are given that the sum of 4 different odd integers is 64 and need to determine the value of the greatest of these integers.

Statement One Alone:

The integers are consecutive odd numbers

Since we know that the integers are consecutive odd integers, we can denote the integers as x, x + 2, x + 4, and x + 6 (notice that the largest integer is x + 6).

Since the sum of these integers is 64, we can create the following equation and determine x:

x + (x + 2) + (x + 4) + (x + 6) = 64

4x + 12 = 64

4x = 52

x = 13

Thus, the largest integer is 13 + 6 = 19.

Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone:

Of these integers, the greatest is 6 more than the least.

Using the information in statement two, we can determine that the four integers are consecutive odd integers. Let’s further elaborate on this idea. If we take any set of four consecutive odd integers, {1, 3, 5, 7}, {9, 11, 13, 15}, or {19, 21, 23, 25}, notice that in ALL CASES the greatest integer in the set is always 6 more than the least integer. In other words, the only way to fit two odd integers between the odd integers n and n + 6 is if the two added odd integers are n + 2 and n +4, thus making them consecutive odd integers. Since we have determined that we have a set of four consecutive odd integers and that their sum is 64, we can determine the value of all the integers in the set, including the value of the greatest one, in the same way we did in statement one. Thus, statement two is also sufficient to answer the question.

Answer: D
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Bunuel : can you just help me understand why from the first statement we are taking x= x + (x + 2) + (x + 4) + (x + 6) = 64. Can't it be x + (x - 2) + (x - 4) + (x - 6) = 64 ..won't the greatest number be different

In my solution the smallest number is x and the greatest is x + 6.

In your case the smallest number is x - 6 and the greatest is x.

In any case the answer is the same.
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(1) The consecutive integers must be (15-2,15,15+2,15+4) so that the sum could be 64,So the greatest number is 19,Sufficient

(2) The range of the integers is 6,within this range to get sum 64 the integers must be (13,15,17,19) as statement (1),Sufficient

Correct Answer D

Target question: What is the value of the greatest of these integers?

Given: The 4 numbers are different odd integers, and their sum is 64.

Statement 1: The integers are consecutive odd numbers
Let x = the first odd integer
So, x + 2 = the 2nd odd integer
So, x + 4 = the 3rd odd integer
So, x + 6 = the 4th odd integer
Since we're told the sum is 64, we can write: x + (x+2) + (x+4) + (x+6) = 64
Since we COULD solve this equation for x, we COULD determine all 4 values, which means we COULD determine the value of the greatest of the 4 odd integers
Of course, we're not going to waste valuable time solving the equation, since our sole goal is to determine whether the statement provides sufficient information.
Since we COULD answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: Of these integers, the greatest is 6 more than the least.
Notice that the 4 CONSECUTIVE integers (from statement 1) can be written as x, x+2, x+4 and x+6
Notice that the biggest number (x+6) is 6 more than the smallest number (x).
Since the 4 odd integers are different, statement 2 is basically telling us that the 4 integers are CONSECUTIVE
So, for the same reason we found statement 1 to be SUFFICIENT, we can also conclude that statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent

Question : What is the greatest of four odd integers which sum up to 64?

Statement 1: The integers are consecutive odd numbers
(a-6)+(a-4)+(a-2)+(a) = 64
i.e. a = 19

SUFFICIENT

STatement 2: Of these integers, the greatest is 6 more than the least
This statement also confirms that the integers are consecutive
(a-6)+(a-4)+(a-2)+(a) = 64
i.e. a = 19

SUFFICIENT

Answer: option D

ritu1009 : Please post DS question in respective forum. You seem to have posted it among PS questions
Bunuel : Please shift the question in the DS forum

There are a lot of explanations on this forum that focus blindly on the math. But remember: the GMAT is a critical-thinking test. Let's talk strategy here. For those of you studying for the GMAT, you will want to internalize strategies that actually minimize the amount of math that needs to be done, making it easier to manage your time. The tactics I will show you here will be useful for numerous questions, not just this one. My solution is going to walk through not just what the answer is, but how to strategically think about it. Ready? Here is the full "GMAT Jujitsu" for this question:

The theme of this entire problem is what I call in my classes “Looking for Leverage.” If a statement initially looks like it is insufficient, look for facts, relationships, and words that you can efficiently use to “squeeze” information out of. Sometimes a single word makes the biggest difference. Watch how we do this with this problem.

Statement #1 tells us that all of the four different odd integers are “consecutive.” This is massive leverage. Consecutive odd integers take the form of \(n\), \(n+2\), \(n+4\), \(n+6\), etc., with \(n\) being the first odd integer. (If you don’t see this immediately, just plug in a concrete value for “\(n\)” to visualize it. For example: \(1\), \(3\), \(5\), \(7\) etc…) So, with this problem, we know that the “sum of 4 different odd integers is 64.” Thus:

\(n + (n+2) + (n+4) + (n+6) = 64\)

We have one equation with a single variable. There is no possibility of multiple possible values (such as with equations containing exponents, absolute values, inequalities, etc.) The real trap of Statement #1 is getting you to think that you actually need to solve for the greatest of the integers (in this case, “\(n+6\)”), instead of stopping as soon as you know you CAN solve. Many people spend too much time on Data Sufficiency questions because they think they need to get to the bitter end. You don’t. As soon as you have enough information to conclude that a statement is either sufficient or insufficient, you can move on. Since we can easily solve for “\(n\)”, we can easily figure out what “\(n+6\)” is. We don’t need to figure out that the four consecutive odd integers are \(13\), \(15\), \(17\), and \(19\). That is just extra work.

Statement #2 similarly requires us to identify small leverage words to squeeze information out of. In this case, statement #2 tells us that the greatest number is \(6\) more than the smallest number. But the question stem also tells us that each odd number is “different.” With only \(6\) separating the greatest odd number from the smallest odd number, the only POSSIBLE situation would be:

\(n + (n+2) + (n+4) + (n+6) = 64\)

And we have already done this analysis. Statement #2 is also sufficient, and the answer is “D”.

Now, let’s look back at this problem from the perspective of strategy. Your job as you study for the GMAT isn't to memorize the solutions to specific questions; it is to internalize strategic patterns that allow you to solve large numbers of questions. This problem can teach us patterns seen throughout the GMAT. First, the structure of this question is what I call “Cousins in Disguise” in my classes. Such problems are not uncommon on the GMAT. “Cousins in Disguise” happen when the two Data Sufficiency statements contain overlapping information, so that either: (1) the information in one statement is completely embedded in the other or (2) combining one statement with information in the question stem leads to the same information given in the other statement. Because of the overlapping information, the answer to “Cousins in Disguise” questions will never be “C”.

This problem also highlights the importance of “Looking for Leverage” in Data Sufficiency questions. (Okay, to be perfectly truthful, the idea of leveraging key details of questions is FUNDAMENTAL to practically every single GMAT question, in every section of the test: Verbal, Quant, IR, and AWA!) And that is thinking like the GMAT.

\(\sum\nolimits_{4\,\,{\rm{different}}\,\,{\rm{odds}}} {\,\, = \,\,\,64\,\,\,\,\left( * \right)}\)

\(? = \,\,{\rm{max}}\,\,{\rm{among}}\,\,{\rm{them}}\)


\(\left( 1 \right)\,\,\,{\rm{consecutive}}\,\,{\rm{and}}\,\,{\rm{sum}}\,\,64\,\,\,\left( {{\rm{from}}\,\,\left( * \right)} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\rm{they}}\,\,{\rm{are}}\,\,{\rm{unique}}!\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\rm{SUFF}}.\,\,\,\,\)

\(\left( 2 \right)\,\,\,{\text{must}}\,\,{\text{be}}\,\,{\text{consecutive}}\,\,\,\left[ {\,\,\underline {2M - 3} \,\,,\,\,2M - 1\,\,,\,\,2M + 1\,\,,\,\,\underline {2M + 3} \,\,} \right]\,\,\,\,\,\, \Rightarrow \,\,\,\,\left( 1 \right)\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\text{SUFF}}.\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

P.S.: the post immediately above is a typical example of a misunderstanding: math is NOT the same as doing calculations or lengthly equations. My course is probably the most mathematically-oriented in the whole PLANET and, even so, my solution above is probably the "less technical" (and probably the less time-consuming) of ALL others presented. Mathematics helps people gain quantitative maturity and THAT´S what the quant section of the GMAT is really about!
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MentorTutoring VeritasKarishma BrentGMATPrepNow chetan2u



I falter at times in deciding what is first step while solving such Qs.
E.g. Is not an odd no represented by 2x+1, where x can be any positive or negative integer including 0.
x can not be fraction.
consecutive odd no: 2x+1, 2x+3, 2x+5

How do we simplify above representation to: a consecutive odd no can be represented by: x, x+2, x+6 ... ?
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Even numbers are represented by 2x
and odd numbers by (2x + 1)
(x can be any integer)

You can write 4 consecutive odd numbers as (2x - 3), (2x - 1), (2x + 1) and (2x + 3).
The calculations will be easy this way.
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Hello, adkikani. If you are given that the integers must be odd, then you could use x to represent the first odd integer, and x, x+2, x+4, and x+6 would give you your four consecutive odd integers (starting with x). If the problem specified even integers, then you could use the same variable and system: x, x+2, x+4, and x+6. To be honest, I did not use an algebraic method for this one. Rather, I figured out the average of the four consecutive integers and determined what the numbers had to be. Let me explain:

Step 1: figure out the average of the four consecutive integers.

\(\frac{64}{4} = 16\)

Step 2: choose the odd integers closest to 16, the ones that "flank" it, that fit the average.

\(\frac{(15+17)}{2}=16\)

\(\frac{(13+19)}{2}=16\)

The integers must be 13, 15, 17, and 19. Which statements lead me to the same conclusion? Statement (1) is SUFFICIENT. The only four consecutive odd integers that will sum to 64 are the ones we have already determined. The answer must be (A) or (D). Statement (2) is SUFFICIENT. We already know that the greatest integer is going to be 6 more than the least, whether we use the algebraic method from earlier or the one outlined above.

I hope that helps. Thank you for tagging me.

- Andrew
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This mothod can help you to find the right answer faster !

Hope it helps

Video solution from Quant Reasoning:
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Answer: Option D

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Statement 1
x + x +2 + x + 4 + x +6 = 64
4x+ 12 = 64
x =13 = > Last value = 13+6 = 19

Statement 2
first + last / 2 = Av
(x + x +6 )/2 = 64 / 4
2x +6 =32
x = 13 = >Last value= 13+6 = 19

INTERESTING!

The answer would actually be the same, even if the odd integers do not have to be different. The least and greatest have to be 13 and 19, respectively.

For instance, with Statement 2:

If the least was 11, the greatest must be 17. And the statement restricts us from having any integers outside of this range. Even if the remaining two integers are maximized to be 17 also, the sum will be smaller than 64.
\(11+17+17+17=62\)

Similarly, if the least was 15, the greatest must be 21. And the statement restricts us from having any integers outside of this range. Even if the remaining two integers are minimized to be 15 also, the sum will be greater than 64.
\(15+15+15+21=66\)

Hi Bunuel,

Might be a silly question, but when should we represent odd numbers as x,x+2,x+4 and when should they be represented as 2x+1, 2x+3, 2x+5. Or do the two representations make no diff? Thanks

You can express odd integers as ..., 2k - 1, 2k + 1, 2k + 3, ... for some integer k. For example, for k = -4, we get: ..., -9, -7, -5, ...
Or you can express odd integers as ..., n - 2, n, n + 2, n + 4, ... for some ODD integer n. For example, for n = 7, we get: ..., 5, 7, 9, 11, ...

It depends on a problem at hand what would be more suitable way of expressing.
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