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# The sum of n consecutive positive integers is 45

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Re: Good set of DS 3 [#permalink]

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16 Sep 2010, 11:30
Thanx Bunuel!Some good questions

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Re: Good set of DS 3 [#permalink]

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19 Dec 2010, 17:08
yangsta8 wrote:
Bunuel wrote:
4. Is y – x positive?
(1) y > 0
(2) x = 1 – y

Statement 1) y>0 Not suff, X could be anything larger or smaller than X.
Statement 2) x=1-y
x+y=1
Let x=3 and y=-2 then y-x < 0.
But if x=1/4 and y=3/4 then y-x >0
Not suff.

1 and 2 together)
From the example above we have:
if x=1/4 and y=3/4 then y-x >0
but if we flip it around:
if x=3/4 and y=1/4 then y-x <0
not suff.

ANS = E

I selected that both stmts were sufficient together because the examples I chose worked both ways so if x=-2 y=3 you get 3-(-2)=5 or if y=2 than x = (-1) so 2-(-1)=3. Can the insufficiency only be seen with fractional numbers? Thanks.

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Re: Good set of DS 3 [#permalink]

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19 Dec 2010, 17:33
gettinit wrote:
yangsta8 wrote:
Bunuel wrote:
4. Is y – x positive?
(1) y > 0
(2) x = 1 – y

Statement 1) y>0 Not suff, X could be anything larger or smaller than X.
Statement 2) x=1-y
x+y=1
Let x=3 and y=-2 then y-x < 0.
But if x=1/4 and y=3/4 then y-x >0
Not suff.

1 and 2 together)
From the example above we have:
if x=1/4 and y=3/4 then y-x >0
but if we flip it around:
if x=3/4 and y=1/4 then y-x <0
not suff.

ANS = E

I selected that both stmts were sufficient together because the examples I chose worked both ways so if x=-2 y=3 you get 3-(-2)=5 or if y=2 than x = (-1) so 2-(-1)=3. Can the insufficiency only be seen with fractional numbers? Thanks.

Yes, in order to get NO answer you should choose values of x and y from (0, 1), note that this range can give you the YES answer as well.

Is $$y-x>0$$? --> is $$y>x$$?

(1) y > 0, not sufficient as no info about $$x$$.
(2) x = 1 - y --> $$x+y=1$$ --> the sum of 2 numbers equal to 1 --> we can not say which one is greater. Not sufficient.

(1)+(2) $$x+y=1$$ and $$y>0$$, still can not determine which one is greater: if $$y=0.1>0$$ and $$x=0.9$$ then $$y<x$$ but if $$y=0.9>0$$ and $$x=0.1$$ then $$y>x$$. Not sufficient.

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Re: Good set of DS 3 [#permalink]

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23 Dec 2010, 19:07
thanks Bunuel should have simplified stmt 2 better to understand the fractional solutions. Thank you sir.

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Re: Good set of DS 3 [#permalink]

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31 Jan 2011, 06:33
Hi Bunuel

For question # 8, please explain how this is true :

either 4y=32 y=8 x=-3, xy=-24 OR -4y=32

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Re: Good set of DS 3 [#permalink]

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31 Jan 2011, 06:48
subhashghosh wrote:
Hi Bunuel

For question # 8, please explain how this is true :

either 4y=32 y=8 x=-3, xy=-24 OR -4y=32

Regards,
Subhash

If x and y are non-zero integers and |x| + |y| = 32, what is xy?

(1) $$-4x-12y=0$$ --> $$x=-3y$$ --> $$x$$ and $$y$$ have opposite signs.

So either: $$|x|=x$$ and $$|y|=-y$$ --> in this case $$|x|+|y|=x-y=-3y-y=-4y=32$$: $$y=-8$$, $$x=24$$, $$xy=-24*8$$;

OR: $$|x|=-x$$ and $$|y|=y$$ --> $$|x|+|y|=-x+y=3y+y=4y=32$$ --> $$y=8$$ and $$x=-24$$ --> $$xy=-24*8$$, the same answer.

Sufficient.

(2) $$|x| - |y| = 16$$. Sum this one with th equations given in the stem --> $$2|x|=48$$ --> $$|x|=24$$, $$|y|=8$$. $$xy=-24*8$$ (x and y have opposite sign) or $$xy=24*8$$ (x and y have the same sign). Multiple choices. Not sufficient.

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Re: Good set of DS 3 [#permalink]

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31 Jan 2011, 19:10
Thanks a lot, I can understand it fully now.
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Re: Good set of DS 3 [#permalink]

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23 Feb 2011, 01:13
For question number 3, consider this:

1st number = 10w+x and 2nd number = 10c+x so three digit number = 100*(wc)+10*(w+c)+x^2

In 1, We are given, wc=x*(w+c)=x^2

or x*(w+c-x) = 0 implying either x=0 or w+c-x = 0
Since it is given that x is non-zero, we have w+c-x = 0 so 1 is sufficient
w+c and x being odd just says that w+c-x is even number, so insufficient

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Re: Good set of DS 3 [#permalink]

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07 Aug 2011, 11:53
Quote:
If a and b are integers, and a not= b, is |a|b > 0?
(1) |a^b| > 0
(2) |a|^b is a non-zero integer.

is true when and does not equal to zero.

(1) --> does not equal to zero, but we don't know about , it can be any value, positive or negative. Not sufficient.

(2) is a non-zero integer --> can be 1 and any integer, positive or negative. Not sufficient.

(1)+(2) If a=1 and b=2, then |a|b > 0, but if a=1 and b=-2, then |a|b <0. Not sufficient.

Thanks for the explanation, Bunuel..
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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20 Apr 2012, 21:10
Bunuel wrote:
Please find below new set of DS problems:

TIP: many of these problems act in GMAT zone, so beware of ZIP trap.

1. The sum of n consecutive positive integers is 45. What is the value of n?
(1) n is even
(2) n < 9

Bunuel, Is this approach correct?

sum of n consecutive +ve integers = $$\frac{n(n+1)}{2}$$

so, $$\frac{n(n+1)}{2}$$ = 45
n(n+1) = 90
$$n^{2}+ n - 90 = 0$$
solving this equation gives n= -10 or 9

Both A and B doesn't help. Hence E.
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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20 Apr 2012, 21:16
ENAFEX wrote:
Bunuel wrote:
Please find below new set of DS problems:

TIP: many of these problems act in GMAT zone, so beware of ZIP trap.

1. The sum of n consecutive positive integers is 45. What is the value of n?
(1) n is even
(2) n < 9

Bunuel, Is this approach correct?

sum of n consecutive +ve integers = $$\frac{n(n+1)}{2}$$

so, $$\frac{n(n+1)}{2}$$ = 45
n(n+1) = 90
$$n^{2}+ n - 90 = 0$$
solving this equation gives n= -10 or 9

Both A and B doesn't help. Hence E.

Bunuel, Ignore my request. I found out the mistake. The question does not say sum of first consecutive integers, so, I cant use this formula.
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Re: Good set of DS 3 [#permalink]

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02 May 2012, 08:11
Hi Bunuel/Karishma,
Request you to clarify the below colored part. What difference does this makes in the question?
Thanks
H
Bunuel wrote:

7. If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value of c?
(1) d = 3
(2) b = 6

Note this part: "for all values of x"
So, it must be true for x=0 --> c=d^2 --> b=2d
(1) d = 3 --> c=9 Sufficient
(2) b = 6 --> b=2d, d=3 --> c=9 Sufficient

.

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Re: The sum of n consecutive positive integers is 45 [#permalink]

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29 Jul 2012, 04:31
1. The sum of n consecutive positive integers is 45. What is the value of n?
(1) n is even
(2) n < 9

Cant we use the formula for sum of n consecutive +ve integers (n)*(n+1)/2 = 45
n^2 +n -90 = 0
n=9 or -10
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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29 Jul 2012, 05:02
dvinoth86 wrote:
1. The sum of n consecutive positive integers is 45. What is the value of n?
(1) n is even
(2) n < 9

Cant we use the formula for sum of n consecutive +ve integers (n)*(n+1)/2 = 45.

[/color]
n^2 +n -90 = 0
n=9 or -10

Cant we use the formula for sum of n consecutive +ve integers (n)*(n+1)/2 = 45.

This formula gives the sum of the first n consecutive positive integers: 1,2 3, ..., n.
Nowhere is stated that our sequence starts with 1.
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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29 Jul 2012, 05:10
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dvinoth86 wrote:
1. The sum of n consecutive positive integers is 45. What is the value of n?
(1) n is even
(2) n < 9

Cant we use the formula for sum of n consecutive +ve integers (n)*(n+1)/2 = 45
n^2 +n -90 = 0
n=9 or -10

As stated in the previous post $$\frac{n(n+1)}{2}$$ gives the sum of first $$n$$ positive integers: $$1+2+3+...+n=\frac{n(n+1)}{2}$$ and we cannot use that formula since we are not told that we have this case. Check this for more: math-number-theory-88376.html (Evenly spaced set chapter).

Solution of this problem is as follows:

The sum of n consecutive positive integers is 45. What is the value of n?

(1) n is even --> n can be 2: 22+23=45. But it also can be 6 --> x+(x+1)+(x+2)+(x+3)+(x+4)+(x+5)=45 --> x=5. At least two values of n are possible. Not sufficient.

(2) n<9 --> the above example is also valid for this statement, hence not sufficient.

(1)+(2) Still at least two values of n are possible. Not sufficient.

Hope it's clear.
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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29 Jul 2012, 05:32
Q1

The sum of n consecutive integers is either a multiple of the middle term, in case n is odd, or a multiple of the sum of the two middle terms, in case n is even.

(1) Since n is even and 45 is odd, we must have an odd number of pairs in our sequence, such that the sum of each pair is a factor of 45, and it is the same as the sum of the two middle terms.
For example, we can have just one pair (22, 23) - or three pairs, each sum being 45/3 = 15 - 5, 6, 7, 8, 9, 10, with 5+10=6+9=7+8=15.
So, (1) is not sufficient.

(2) From what we have seen above, (2) is not sufficient either.
Just as an example, if n is odd and less than 9, we can have the sequence 14, 15, 16.

(1) and (2) taken together is obviously not sufficient.

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Re: The sum of n consecutive positive integers is 45 [#permalink]

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29 Jul 2012, 05:38
Q7

Since the given equality must hold for any value of x, if we substitute x = 0, we obtain $$c=d^2$$.

Then, we can immediately see that (1) alone is sufficient, but (2) is not.

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Re: The sum of n consecutive positive integers is 45 [#permalink]

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29 Jul 2012, 05:49
Q2. Is a product of three integers XYZ a prime?
(1) X=-Y
(2) Z=1

(1) We can take X=1, then Y=-1, and taking Z any non-negative integer will get a non-prime, as the product will be either negative or 0.
But if we take X=1, Y=-1 and for example Z=-2, than XYZ=2, which is a prime.
Therefore, (1) is not sufficient.

(2) If Z=1, we can take Y=1 as well, and then either X is a prime or not, so the final product XYZ=X, can be or not a prime.
Not sufficient.

(1) and (2) together: If Z=1 and X=-Y, then $$XYZ=-X^2$$, which can be either 0 (when X=0) or a negative integer.
In either case, we won't get a prime number (by definition, a prime must be a positive integer).
So, the answer is a definite NO, therefore sufficient.

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Re: The sum of n consecutive positive integers is 45 [#permalink]

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29 Jul 2012, 06:18
Q3. Multiplication of the two digit numbers wx and cx, where w,x and c are unique non-zero digits, the product is a three digit number. What is w+c-x?
(1) The three digits of the product are all the same and different from w c and x.
(2) x and w+c are odd numbers.

(1) If the product is a three digit number, all digits identical, then the number must be of the form AAA=A*111=A*3*37, where A is a non-zero digit.
It means that x = 7, and 3A must be a two digit number, which also ends in 7. It follows that 3A = 27, so the numbers are 37 and 27, 37 * 27 = 999,
and w+c-x=5-2, sufficient (it doesn't matter who is w and who is c).

(2) Since w+c is odd, one of them must be even and the other one odd. Fro example, we can take w=2, c=1, x=3 or w=3, c=2, x=1.
23*13=299, 31*21=651, are both three digit numbers, and w+c-x is 0 and 4, respectively.
Not sufficient.

Remark: In the body of the question, w,x and c are unique non-zero digits, shouldn't be "distinct non-zero digits"?
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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29 Jul 2012, 06:27
Q4. Is y – x positive?
(1) y > 0
(2) x = 1 – y

The question we are asked is in fact is y>x?

(1) Not sufficient, as we don't know anything about x.
(2) We have x+y=1, again not sufficient. We can have x=y=0.5, or x=0.25 < y=0.75, or x=0.75 > y=0.25

(1) and (2) not sufficient, we can use the examples from the analysis for (2) above.

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Re: The sum of n consecutive positive integers is 45   [#permalink] 29 Jul 2012, 06:27

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