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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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26 Nov 2012, 01:40
Amateur wrote:
Bunuel wrote:
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.

i did not understand the explanation you gave..... a prime is a number which is divisible by 1 and itself right? if x,y,z are three integers..... and for it to be prime.... two for those three integers should be 1 or -1 or 1,-1.... so the third one be prime number or negative prime number....
(1) says two of them are equal in magnitude... so z can be -p to be prime or negative composite number or positive non prime in either case not sufficient...
(2) z=1 nothing said about x,y..... not sufficient

(1) + (2)
product will be a positive or negative composite number or 1..... so not a prime which is sufficient....
am i thinking correctly?

We have that $$x=-y$$ and $$z=1$$, thus $$xyz=-x^2$$. Now, $$-x^2\leq{0}$$, thus it cannot be a prime number (only positive numbers are primes).

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

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06 Jan 2013, 03:24
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.
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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07 Jan 2013, 03:20
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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.

If $$x=-1$$, $$y=1$$, $$z=-7$$, then $$xyz=(-1)*1*(-7)=7=prime$$.

Check here: the-sum-of-n-consecutive-positive-integers-is-85413-20.html#p667155 and here: the-sum-of-n-consecutive-positive-integers-is-85413.html#p640133

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

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13 Sep 2013, 06:32
Hi Bunnel,

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

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13 Sep 2013, 07:47
GMAT40 wrote:
Hi Bunnel,

(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.

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

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13 Sep 2013, 09:13
Bunuel wrote:
GMAT40 wrote:
Hi Bunnel,

(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.

My question was how did you arrive at 37 * K
and how did you rule out 2, 3, 7

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

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22 Nov 2013, 09:20
I had a problem with number 8 and 9
8
Why is statement 2 not sufficient? I mean |x| means positive x so cant we then arrange it as |x|-|y|=16
Same as x-y=16 then use substitution method with the equation |x|+|y|=32 above?
9
You mentioned that “when odd number n is doubleb, 2n has twice as many factors as n”
Is this always the case? Let’s say our odd number is 15 ,it has four factors 5 ,1,15and 3.when doubled it becomes 30.30 has 30,1,2,3,5 factors. Just one more factor than 15.
My understanding for YES/NO DS question is that a statement is sufficient only if it satisfies the question always.
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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26 Nov 2013, 07:35
mumbijoh wrote:
I had a problem with number 8 and 9
8
Why is statement 2 not sufficient? I mean |x| means positive x so cant we then arrange it as |x|-|y|=16
Same as x-y=16 then use substitution method with the equation |x|+|y|=32 above?
9
You mentioned that “when odd number n is doubleb, 2n has twice as many factors as n”
Is this always the case? Let’s say our odd number is 15 ,it has four factors 5 ,1,15and 3.when doubled it becomes 30.30 has 30,1,2,3,5 factors. Just one more factor than 15.
My understanding for YES/NO DS question is that a statement is sufficient only if it satisfies the question always.

Question 8 is discussed here: if-x-and-y-are-non-zero-integers-and-x-y-32-what-is-128845.html

Question 9 is discussed here: is-the-integer-n-odd-1-n-is-divisible-by-3-2-2n-is-91399.html

15 has 4 factors: 1, 3, 5, and 15.
30 has 8 factors: 1, 2, 3, 5, 6, 10, 15, and 30.

Hope this helps.
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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18 Apr 2014, 06:16
Bunuel wrote:
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.

am I missing anything?
it does not say that w x and c are positive, does it?
w= 3, c= 2 and x= 7
37*27 = 999, here w+c -7 = -2

but we also can have
w= -3 and c= -2 and x= 7
-37*-27 = 999, here w+c - 7 = -12

Both of these sets satisfy both the conditions , hence I am getting E,
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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18 Apr 2014, 10:12
qlx wrote:
Bunuel wrote:
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.

am I missing anything?
it does not say that w x and c are positive, does it?
w= 3, c= 2 and x= 7
37*27 = 999, here w+c -7 = -2

but we also can have
w= -3 and c= -2 and x= 7
-37*-27 = 999, here w+c - 7 = -12

Both of these sets satisfy both the conditions , hence I am getting E,

w, x and c are unique non-zero digits of the two digit numbers wx and cx means that w, x, and c are 1, 2, 3, 4, 5, 6, 7, 8, or 9.

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

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24 Apr 2014, 02:08
Bunuel wrote:
qlx wrote:
Bunuel wrote:
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.

am I missing anything?
it does not say that w x and c are positive, does it?
w= 3, c= 2 and x= 7
37*27 = 999, here w+c -7 = -2

but we also can have
w= -3 and c= -2 and x= 7
-37*-27 = 999, here w+c - 7 = -12

Both of these sets satisfy both the conditions , hence I am getting E,

w, x and c are unique non-zero digits of the two digit numbers wx and cx means that w, x, and c are 1, 2, 3, 4, 5, 6, 7, 8, or 9.

Hope it's clear.

Sorry for the delayed post, but couldn't it be better if it was mentioned that w, c, x are non zero positive integers?

"w, x and c are unique non-zero digits of the two digit numbers wx and cx" - From this statement how can we deduce or assume that the 2 digit number cannot be negative ?

As of now one could argue wx = -37 and cx = -27 is also a possibility.
here I have taken w = -3 and x= 7 and c= -2

Hope you agree.

Thank you and appreciate your help.
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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24 Apr 2014, 02:14
qlx wrote:
Bunuel wrote:
qlx wrote:
w, x and c are unique non-zero digits of the two digit numbers wx and cx means that w, x, and c are 1, 2, 3, 4, 5, 6, 7, 8, or 9.

Hope it's clear.

Sorry for the delayed post, but couldn't it be better if it was mentioned that w, c, x are non zero positive integers?

"w, x and c are unique non-zero digits of the two digit numbers wx and cx" - From this statement how can we deduce or assume that the 2 digit number cannot be negative ?

As of now one could argue wx = -37 and cx = -27 is also a possibility.
here I have taken w = -3 and x= 7 and c= -2

Hope you agree.

Thank you and appreciate your help.

Two-digit numbers wx means positive integer, where w is the tens digit and x is the units digit: 10, 11, 12, ..., 99.

Two-digit numbers -wx means negative integer, where w is the tens digit and x is the units digit: -10, -11, -12, ..., -99.
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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24 Apr 2014, 02:44
Bunuel wrote:
Two-digit numbers wx means positive integer, where w is the tens digit and x is the units digit: 10, 11, 12, ..., 99.

Two-digit numbers -wx means negative integer, where w is the tens digit and x is the units digit: -10, -11, -12, ..., -99.

Thank you I finally got what you are saying.

I think a variable representing an integer can be positive or negative ( or zero) , a variable without a minus sign does not mean that the variable is positive does it .

using the same logic
if a particular sum says

if a and b are integers what is b ?

1) a+b = 1
2) a= 2

From here can we say that a is positive ,because it is not -a ?
so a means = a is a positive integer
and -a means = a is a negative integer

In the same way when it says wx is a two digit integer , how can we say because it is not -wx hence wx is positive.

Last edited by qlx on 24 Apr 2014, 03:09, edited 1 time in total.
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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24 Apr 2014, 02:57
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qlx wrote:
Bunuel wrote:
Two-digit numbers wx means positive integer, where w is the tens digit and x is the units digit: 10, 11, 12, ..., 99.

Two-digit numbers -wx means negative integer, where w is the tens digit and x is the units digit: -10, -11, -12, ..., -99.

Thank you I finally got what you are saying.

I think a variable representing an integer can be positive or negative , a variable without a minus sign does not mean that the variable is positive does it .

using the same logic
if a particular sum says

if a and b are integers what is b ?

1) a+b = 1
2) a= 2

From here can we say that a is positive ,because it is not -a ?
so a means = a is a positive integer
and -a means = a is a negative integer

In the same way when it says wx is a two digit integer , how can we say because it is not -wx hence wx is positive.

a is an integer does not mean that a is positive but a is a digit means that a is positive: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.

Sorry cannot explain any better.
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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24 Apr 2014, 03:27
Bunuel wrote:
qlx wrote:
Bunuel wrote:
Two-digit numbers wx means positive integer, where w is the tens digit and x is the units digit: 10, 11, 12, ..., 99.

Two-digit numbers -wx means negative integer, where w is the tens digit and x is the units digit: -10, -11, -12, ..., -99.

Thank you I finally got what you are saying.

I think a variable representing an integer can be positive or negative , a variable without a minus sign does not mean that the variable is positive does it .

using the same logic
if a particular sum says

if a and b are integers what is b ?

1) a+b = 1
2) a= 2

From here can we say that a is positive ,because it is not -a ?
so a means = a is a positive integer
and -a means = a is a negative integer

In the same way when it says wx is a two digit integer , how can we say because it is not -wx hence wx is positive.

a is an integer does not mean that a is positive but a is a digit means that a is positive: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.

Sorry cannot explain any better.

Thank you for your explanation ,

So the terms " Integer" and " Digit " makes all the difference ?They do not mean the same thing, right?

Integer =both negative , positive or zero
Digit = only 0 or positive.

I thought ( -2) was a digit too! a negative digit? Can we not have a negative digit?

Or is the Digit term in GMAT only used for non negative integers.

I think this is important info in terms of Gmat.

Really appreciate your help.Thank you so much .
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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01 May 2014, 21:29
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qlx wrote:
So the terms " Integer" and " Digit " makes all the difference ?They do not mean the same thing, right?

Integer =both negative , positive or zero
Digit = only 0 or positive.

I thought ( -2) was a digit too! a negative digit? Can we not have a negative digit?

Or is the Digit term in GMAT only used for non negative integers.

I think this is important info in terms of Gmat.

Really appreciate your help.Thank you so much .

Responding to a pm: Here goes my honest opinion you asked for!

An integer can be positive, negative or 0.

A single digit, on the other hand always implies a positive single digit from 0 to 9.
Given ab where a and b are single digits, ab is a positive integer.

In the question, you have "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".
This implies that w, c and x are distinct digits from 1 to 9 and wx and cx are positive integers.

The author did not forget to mention "positive digits". He/she did not need to mention it because digits imply positive digits only.

Also, in case of confusion, you can always search on Google. Say, put "digit in Math" and check out the various write ups. If there are multiple usages, the net will tell you that too.
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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04 May 2014, 06:15
Hi bunnel

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..... how is x = -3, when y = 8?... isnt x=-3y??
(2) Multiple choices. Not sufficient.
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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04 May 2014, 06:47
nandinigaur wrote:
Hi bunnel

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..... how is x = -3, when y = 8?... isnt x=-3y??
(2) Multiple choices. Not sufficient.

There was a typo. Correct solution is below:

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

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04 May 2014, 09:21
Hi bunnel

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

in this qs...
i have got confused...
for st 1:

1 got several values of n therefore insufficient.

but in statement 2 what does your explanation mean by: 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.? how do you know its the some of exactly 9 consecutive numbers?
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Re: The sum of n consecutive positive integers is 45 [#permalink]

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04 May 2014, 10:01
nandinigaur wrote:
Hi bunnel

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

in this qs...
i have got confused...
for st 1:

1 got several values of n therefore insufficient.

but in statement 2 what does your explanation mean by: 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.? how do you know its the some of exactly 9 consecutive numbers?

Because the least sum of 10 consecutive positive integers is 1+2+3+4+5+6+7+8+9+10=55.
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