The sum of n consecutive positive integers is 45

Thats cool.

You can say PINZF (or better) trap as well:

P = positive
I = integer
N = negative
Z = zero
F = fraction

Did you say "consecutive positive integers"? not only "consecutive integers"?

Agree with B if "consecutive positive integers". If only "consecutive integers", B is correct.

Seems you said "consecutive positive integers"!

sum of n integers = (n*(n+1))/2

(n*(n+1))/2 = 45 which yields n = 9 could be the only answer

Stmt 1 and 2 (D) are both sufficient.

Does GMAT assume that we would not indulge in any formula? Just curious

I like the concept of ZIP trap

check for 0,-ve and fractions.

This is not correct.

First of all the formula you referring n(1+n)/2 is the formula for counting the sum of n FIRST integers (meaning that starting from 1). Question stem does not give us that information: "The sum of n consecutive positive integers is 45." You can not assume anything.

The formula for counting the sum of consecutive integers (not necessarily first n integers), which is in fact sum of AP (arithmetic progression) is:
Sn=n*(a1+an)/2 or Sn=n*(2a1+d(n-1))/2, you can substitute d=1, as the numbers are consecutive.

If you further substitute a1 (the first term of the progression) with 1 you'll get exactly the formula you wrote: Sn=n(2+n-1)/2=n(n+1)/2

Second, you can find the n consecutive odd positive integers (n=odd) to total 45:
n=3 --> 14, 15, 16
n=5 --> 7, 8, 9, 10, 11.

Hope it's clear.

8 is also not possible.
........................................................................

1. wx * cx = aaa (aaa can only be 111 (for 9), 444 (for 2), 666 (for 8), and 999 (for 3 and 7)). 222, 333, 555, 777, and 888 are not possible as noneo f the unit difgits of the 3 digit numbers is square of an integer.

wx * cx = 111 = 3x37. Not possible as there is only one 2 digit number. Needs 2 two digit numbers.
wx * cx = 444 = 2x3x37 = 6x37. Not possible as there is only one 2 digit number. Needs 2 two digit numbers.
wx * cx = 666 = 2x3x3x37 = 18x37. Not possible as the unit digit of these 2 digit numbers are not the same.
wx * cx = 999 = 3x3x3x37 = 27x37. Possible as the unit digit of these 2 digit numbers are same.


2. is self explanatory..

please explain 2nd Q.


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

Hi Bunuel.....

Would appreciate if you could explain this in detail as I am confused...

Thanks!
JT

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.

\(|a|b > 0\) is true when \(b>0\) and \(a\) does not equal to zero.

(1) \(|a^b| > 0\) --> \(a\) does not equal to zero, but we don't know about \(b\), it can be any value, positive or negative. Not sufficient.

(2) \(|a|^b\) is a non-zero integer --> \(a\) can be 1 and \(b\) 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.

Answer: E.

This is the easiest way to solve this problem. In my previous solution I just tried to show what each statement means algebraically. Please tell which part needs more clarification.

Not able to get why do you say this (marked in red)... I know that \(|a|\) would always give a positive value, but why cant we consider \(a\) as 2 or 3...? Any reason for this?

OK. For statement (2): First of all I'm not saying that \(a\) can ONLY be 1. I'm saying that it MAY be 1 and in this case \(b\) can take any value statement (2): "\(|a|^b\) is a non-zero integer " to hold true. So \(b\) can take positive as well as negative values, hence \(|a|b > 0\) may or may not be true. That's why (2) is not sufficient.


To elaborate further: we are told that \(|a|^b\) is A. not zero and B. it's an integer.

A. Means that \(a\) is not zero, as the only way \(|a|^b\) to be zero, is \(a\) to be zero.

B. \(|a|^b\) is an integer (non=zero). What does that mean?

When \(a\) is any integer so that \(|a|>1\) (eg 2, -2, 3, -3, ...), \(b\) can take any value but negative. Example: \(a=-3\) --> \(|-3|=3\), \(3\) in integer power (given \(a\) and \(b\) are integers) to be an integer, power (\(b\)), must be positive or zero, as when \(b\) is negative, let's say \(-2\), we'll have: \(3^{-2}=\frac{1}{3^2}\) which is not integer.

But when \(|a|=1\) (\(1\) or \(-1\)), then \(b\) can take ANY integer value as 1 in any power equals to 1 which is integer.

So from statement (2) we'll have:

\(a\) does not equals to 0. Plus, \(b\) can be ANY integer (positive or negative or zero) when \(|a|=1\) and \(b\) must be positive or zero when \(|a|>1\).

Hope it's clear.

jeeteshsingh,

To further clarify what Bunuel was saying. If b was negative and a was not 1 or -1, then |a|^b would be a fraction due to the properties of negative exponents. If |a|=1 (meaning a=1 or a=-1) then the denominator of the fraction would be some power of 1 which is also one. In this case the fraction would be 1 over 1 which is again 1 and satisfies the conditions.

Also, Bunuel, for the second portion I noticed that you didn't include in your listed possibilities that for |a|^b is a non zero integer, b=0 is possible as it would create an answer of 1 for any value of a including 0.

Welcome to the Gmat Club.

Your logic is correct. Though b=0 IS included as one of the possibilities. See the bolded part from the explanation:



You also mention \(0^0\) issue, this not the case for statement 2 (as \(a\) cannot be zero), but still as you brought this up:

\(0^0\), in some sources equals to 1, some mathematicians say it's undefined. Anyway you won't need this for GMAT:

"During the past decade, mathematicians argued extensively about the value of 0^0. Some answer that 0^0 = 1, while others answer that 0^0 is undefined. In the unlikely event that this question appears in some format or is a required intermediary calculation, the correct answer is more likely that 0^0 = 1."
https://www.platinumgmat.com/gmat_study_ ... ial_powers

and:
"Note: the case of 0^0 is not tested on the GMAT."
https://www.manhattangmat.com/np-exponents.cfm

Hi,Can anyone explain me how |a| is positive?We generally look at the sign of any number before deciding its abs value..Right?But how come that is not followed here?
Generally we take
|x|=-x (x <0)
|x|=x (x>0) but y is |a| taken positive directly?In the question,it is just mentioned as an integer

Explanation plz..

Seems that you need to brush up on absolute value. Please check Walker's post on Absolute Value at: math-absolute-value-modulus-86462.html

\(|x|=-x\), when \(x<0\) and \(|x|=x\) when \(x>0\) --> CORRECT.

But when \(x=negative<0\) then \(|x|=-x=-(negative)=positive\) and when \(x=positive>0\) then \(|x|=x=positive\), so in any case when \(x\) is either positive or negative \(|x|\) is still positive. There is one more case though: when \(x=0\) then \(|x|=0\).

So generally we can say that absolute value of an expression is alway non-negative: \(|some \ expression|\geq{0}\) --> \(|x|\geq{0}\).

As for this particular question: see my posts on it on pages 1, 2, and 3.

Hope it helps.

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.

Answer: E.

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.

Answer: A.