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The sum of n consecutive positive integers is 45 [#permalink]
16 Oct 2009, 18:59

15

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Expert's post

00:00

A

B

C

D

E

Difficulty:

5% (low)

Question Stats:

17% (02:11) correct
82% (01:17) wrong based on 28 sessions

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

2. Is a product of three integers XYZ a prime? (1) X=-Y (2) Z=1

3. 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.

4. Is y – x positive? (1) y > 0 (2) x = 1 – y

5. 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

6. If M and N are integers, is (10^M + N)/3 an integer? 1. N = 5 2. MN is even

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

8. If x and y are non-zero integers and |x| + |y| = 32, what is xy? (1) -4x - 12y = 0 (2) |x| - |y| = 16

9. Is the integer n odd (1) n is divisible by 3 (2) 2n is divisible by twice as many positive integers as n

10. The sum of n consecutive positive integers is 45. What is the value of n? (1) n is odd (2) n >= 9

Please share your way of thinking, not only post the answers.

Re: Good set of DS 3 [#permalink]
17 Oct 2009, 16:44

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Expert's post

Yes and about the ZIP trap:

GMAT likes to act in the zone -1<=x<=1. So I always ask myself:

Did I assumed, with no ground for it, that variable can not be Zero? Check 0! Did I assumed, with no ground for it, that variable is an Integer? Check fractions! Did I assumed, with no ground for it, that variable is Positive? Check negative values!

I called it ZIP trap. Helps me a lot especially with number property problems.
_________________

Re: Good set of DS 3 [#permalink]
17 Oct 2009, 16:29

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Expert's post

ANSWERS:

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

(1) n=2 --> 22+23=45, n=4 --> n=6 x1+(x1+1)+(x1+2)+(x1+3)+(x1+4)+(x1+5)=45 x1=5. At least two options for n. Not sufficient. (2) n<9 same thing not sufficient. (1)+(2) No new info. Not sufficient.

Answer: E.

2. Is a product of three integers XYZ a prime? (1) X=-Y (2) Z=1

(1) x=-y --> for xyz to be a prime z must be -p AND x=-y shouldn't be zero. Not sufficient. (2) z=1 --> Not sufficient. (1)+(2) x=-y and z=1 --> x and y can be zero, xyz=0 not prime OR xyz is negative, so not prime. In either case we know xyz not prime.

Answer: C

3. 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) wx+cx=aaa (111, 222, ... 999=37*k) --> As x is the units digit in both numbers, a can be 1,4,6 or 9 (2,3,7 out because x^2 can not end with 2,3, or 7. 5 is out because in that case x also should be 5 and we know that x and a are distinct numbers). 1 is also out because 111=37*3 and we need 2 two digit numbers. 444=37*12 no good we need units digit to be the same. 666=37*18 no good we need units digit to be the same. 999=37*27 is the only possibility all digits are distinct except the unit digits of multiples. Sufficient (2) x and w+c are odd numbers. Number of choices: 13 and 23 or 19 and 29 and w+c-x is the different even number.

Answer: A.

4. Is y – x positive? (1) y > 0 (2) x = 1 – y

Easy one even if y>0 and x+y=1, we can find the x,y when y-x>0 and y-x<0 Answer: E.

5. 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

This is tricky |a|b > 0 to hold true: a#0 and b>0.

(1) |a^b|>0 only says that a#0, because only way |a^b| not to be positive is when a=0. Not sufficient. NOTE having absolute value of variable |a|, doesn't mean it's positive. It's not negative --> |a|>=0

(2) |a|^b is a non-zero integer. What is the difference between (1) and (2)? Well this is the tricky part: (2) says that a#0 and plus to this gives us two possibilities as it states that it's integer: A. -1>a>1 (|a|>1), on this case b can be any positive integer: because if b is negative |a|^b can not be integer. OR B. |a|=1 (a=-1 or 1) and b can be any integer, positive or negative. So (2) also gives us two options for b. Not sufficient.

(1)+(2) nothing new: a#0 and two options for b depending on a. Not sufficient.

Answer: E.

6. If M and N are integers, is (10^M + N)/3 an integer? (1) N = 5 (2) MN is even

Note: it's not given that M and N are positive. (1) N=5 --> if M>0 (10^M + N)/3 is an integer ((1+5)/3), if M<0 (10^M + N)/3 is a fraction ((1/10^|M|+5)/3). Not sufficient. (2) MN is even --> one of them or both positive/negative AND one of them or both even. Not sufficient (1)+(2) N=5 MN even --> still M can be negative or positive. Not sufficient.

Answer: E.

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

Answer: D.

8. If x and y are non-zero integers and |x| + |y| = 32, what is xy? (1) -4x - 12y = 0 (2) |x| - |y| = 16

(1) x+3y=0 --> x and y have opposite signs --> either 4y=32 y=8 x=-3, xy=-24 OR -4y=32 y=-8 x=3 xy=24. The same answer. Sufficient. (2) Multiple choices. Not sufficient.

Answer: A.

9. Is the integer n odd (1) n is divisible by 3 (2) 2n is divisible by twice as many positive integers as n

(1) 3 or 6. Clearly not sufficient. (2) TIP: When odd number n is doubled, 2n has twice as many factors as n. Thats because odd number has only odd factors and when we multiply n by two we remain all these odd factors as divisors and adding exactly the same number of even divisors, which are odd*2.

Sufficient.

Answer: B.

10. The sum of n consecutive positive integers is 45. What is the value of n? (1) n is odd (2) n >= 9

Look at the Q 1 we changed even to odd and n<9 to n>=9

(1) not sufficient see Q1. (2) As we have consecutive positive integers max for n is 9: 1+2+3+...+9=45. (If n>9=10 first term must be zero. and we are given that all terms are positive) So only case n=9. Sufficient.

Re: Good set of DS 3 [#permalink]
26 Dec 2009, 08:04

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Expert's post

jan4dday wrote:

please explain 2nd Q.

i want an example where XYZ can be prime using STATEMENT 1 ALONE

First note prime numbers are only positive. (Also note that x, y and z are integers)

Q: xyz=p, is p prime?

(1) x=-y --> p=-x^2z. Let's check when this expression gives a prime number:

Well first of all p to be prime z MUST be negative, as p MUST be positive to be a prime.

Next if x>|1|, (eg |2|, |3|, ...) OR equals to zero, p won't be prime. So x must be equal to |1|.

But it's not enough. We'll have p=-x^2z=-z, so p to be a prime number z must be equal to -prime.

You are asking how using statement (1) p could be a prime: according to above, when |x|=1 and z=-p. eg.: x=-1 --> y=1 --> z=-7 --> p=(-1)*1*(-7)=7, which is prime.

Statement (1) may or may not give the prime number for xyz. Not sufficient.

(2) z=1 --> p=xy. Again for p to be a prime number xy must be >0 (both positive or both negative). Then if x=|prime| and y=|1|, OR y=|prime| and x=|1|, so that xy>0, then xy is a prime number. For any other values or combinations of x and y, p won't be a prime. Not sufficient.

(1)+(2) p=xyz=-x^2 (as x=-y and z=1). -x^2 is never positive, hence p is not a prime. Sufficient.

Re: Good set of DS 3 [#permalink]
16 Oct 2009, 21:55

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Bunuel wrote:

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

2. Is a product of three integers XYZ a prime? (1) X=-Y (2) Z=1

What is the ZIP trap?

Q1) Statement 1) n = 2,4,6 etc n = 2 => x+(x+1)=45 => x=22 (works) n = 4 => x+(x+1)+(x+2)+(x+3)=45 => 4x+6=45 => x=39/4 (doesn't work) n = 6 => Take above equation+(x+4)+(x+5) => 6x+15=45 => x=5 (works) Not suff. Statement 2) n < 9. This is proven insufficient from the working above since both n=2 and n=6 n<9. 1 and 2 together still prove insufficient due to above working.

ANS = E.

Q2) Statement 1) X=-Y This means Z needs to be negative and for XYZ to have a chance of being prime. Z can be anything. Insufficient. Statement 2) Z=1 X and Y could be anything such as 2 and 3 (non prime multiple) or 1 and 2 (prime). Insufficient. 1 and 2 Together) Z = 1. X=-Y 1*Y*(-Y) = -Y^2 which cannot be prime as it is negative.

ANS = C

Edited: Got the right working but wrote E instead of C. I gotta stop doing that

Last edited by yangsta8 on 16 Oct 2009, 22:03, edited 1 time in total.

Re: Good set of DS 3 [#permalink]
16 Oct 2009, 22:13

1

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Bunuel wrote:

6. If M and N are integers, is (10^M + N)/3 an integer? 1. N = 5 2. MN is even

Statement 1) N=5 If M>=0 then it is always divisible by 3. Since the number will always consist of 1, trailing 0's and a 5. Of which the sum of digits =6 which is the rule for divisibility by 3. If M<0 then the equation is not divisble by 3. For example if M=-1. Insufficient

Statement 2) MN is even. Again this means M could still be negative so insufficient. For example M could be -1 and N could be 2 which is not divisible by 3. Or n=5 but m=-2 which is not.

Statements together) Still insuff. m=2 n=5 works. But m=-2 n=5 doesn't work.

Re: Good set of DS 3 [#permalink]
17 Oct 2009, 01:50

1

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Bunuel wrote:

3. 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.

WX x CX = IJK

1.) I,J,K are the same and not equal to W,C or X.

so 3 digit numbers with all digit same are 111,222,...., 999.

basically multiples of 111 (37x3).

so we get 1 number = 37

conditions the second number has to meet = last digit = 7, multiple of 3, double digit.

Re: The sum of n consecutive positive integers is 45 [#permalink]
29 Jul 2012, 06:10

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Expert's post

EvaJager wrote:

Q8. If x and y are non-zero integers and |x| + |y| = 32, what is xy? (1) -4x - 12y = 0 (2) |x| - |y| = 16

(1) From the given equality we get 4x=-12y, or x=-3y, which gives |x|=3|y|. We can deduce that |y|=8, |x|=24, and |x||y|=|xy|=192. Not sufficient, because xy=192 or -192.

(2) Since |x|=|y|+16, we find again that |y|=8, |x|=24, and |x||y|=|xy|=192. Same situation as in (1), not sufficient.

(1) and (2) together cannot help, as seen above.

Answer E

Answer to this question is A, not E.

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

(1) -4x - 12y = 0 --> x+3y=0 --> x=-3y --> x and y have opposite signs --> so either |x|=x and |y|=-y OR |x|=-x and |y|=y --> either |x|+|y|=-x+y=3y+y=4y=32: y=8, x=-24, xy=-24*8OR|x|+|y|=x-y=-3y-y=-4y=32: y=-8, x=24, xy=-24*8, 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.

Re: The sum of n consecutive positive integers is 45 [#permalink]
07 Jan 2013, 03:20

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pashraddha wrote:

Q Is a product of three integers XYZ a prime? (1) X=-Y (2) Z=1

I'm unable to understand why (1) X=-Y is not sufficient to answer the question?

In all cases if (1) X=-Y, XYZ can not be a prime number, whether X, Y being 0 or Z being negative. I may be missing out something very basic, please help.

Re: Good set of DS 3 [#permalink]
16 Oct 2009, 22:03

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.

Re: Good set of DS 3 [#permalink]
16 Oct 2009, 22:41

Bunuel wrote:

9. Is the integer n odd (1) n is divisible by 3 (2) 2n is divisible by twice as many positive integers as n

10. The sum of n consecutive positive integers is 45. What is the value of n? (1) n is odd (2) n >= 9

Q9) Statement 1) N is a multiple of 3. N could be 3 or 6. Insufficient. Statement 2) I am not sure how to prove this except by examples: Example 1:n=9 factors={1,3,9}, 2n=18 factors={1,2,3,6,9,18} N is odd is true. Example 2:n=6 factors={1,2,3,6} 2n=12 factors={1,2,3,4,6,12} Does not have twice as many factors. Example 3: n=3 factors={1,3} 2n=6 factors={1,2,3,6} N is odd is true.

ANS = B

Q10) Statement 1) N is odd. N could be 1. 45 N could also be 3. x+(x+1)+(x+2)=45 => 3x=42 x=14 Insufficient. Statement 2) N>=9 Let n=9. 9x+8+7+6+5+4+3+2+1=45 => 9x+36=45 => 9x=9 x=1 we cannot use n>10 because adding anymore positive integers means sum > 45. Sufficient.