If a is the sum of x consecutive positive integers. b is the

Hi chikki420,

We're told that A is the sum of X CONSECUTIVE positive integers and B is the sum of Y CONSECUTIVE positive integers. We're asked for which of the following values of X and Y is it IMPOSSIBLE that A = B. This question is based on a subtle Number Property rule (and if you know the rule, then the question becomes a whole lot easier to deal with - and even if you don't know it, then you can figure out it with a little bit or Arithmetic). You can certainly TEST THE ANSWERS to prove what's possible and what's not (and I explain how in an earlier post in this thread), but I'm going to jump directly to what you've mentioned in your post.

This question is NOT asking us to prove which four options lead to equal sums; it's asking us for the one option in which equal sums can NEVER occur (and you do not actually need to know the starting values of each group to prove that).

Since we're dealing with CONSECUTIVE POSITIVE INTEGERS, some interesting patterns occur (based on the sum of those integers being either ODD or EVEN). Take a look at answer D....

X = 10 and Y = 4

The sum of 10 consecutive positive integers will ALWAYS be ODD... since there are 5 ODD numbers in that group (try it with any group of 10 consecutive positive integers and you'll see).
The sum of 4 consecutive positive integers will ALWAYS be EVEN... since there are 2 ODD numbers in that group (again, try it and you'll see)

An ODD number will NEVER equal an EVEN number.

So, A can NEVER EQUAL B in this circumstance, so this specific situation is the answer to this question.

GMAT assassins aren't born, they're made,
Rich

For consecutive integers:
Sum= n/2 (2a1+n-1), where a1 is the first term
Average = Median= Sum/n, where n= number of terms in the set
Value of average = median = either an integer (if n is odd) or integer+0.5 (if n is even)

Question stem is asking us to test all the choices and select the one where sum of one set cannot be equal to the sum of the other.
Simply put, Sum of one set cannot result into an average of either an integer (if n is odd) or (integer +0.5) (if n is even) on the other set.

For n=6, Sum S6 = 3(2a1+5)
For n=9, Sum S9 = 9/2 (2a1+8)= 9(a1+4)
For n=10, Sum S10 = 5(2a1+9)

A) For n=2, Median = S6 /2 = odd value /2= Integer.5, always
B) For n=3, Median = S6 /3 = multiple of 3 /3= Integer, always
C) For n=7, Median = S9 /7. Here, a1+4 /7= Integer when a1= (3+ multiples of 7)
D) For n=4, Median = S10 /4 = odd value /4= Integer.25. NOT POSSIBLE TO DERIVE Integer.5
E) For n=7, Median = S10 /7. Here, (2a1+9) /7= Integer when (2a1+9) is an (odd) multiple of 7.

Ans D

gmatophobia your approach?

Given: If a is the sum of x consecutive positive integers. b is the sum of y consecutive positive integers.
Asked: For which of the following values of x and y is it impossible that a = b?

a = xs1 + x(x+1)/2
b = ys2 + y(y+1)/2

A. x = 2; y = 6
a = 2s1 + 3 ; b = 6s2 + 21; 2s1 + 3 = 6s2 + 21; 2s1 - 6s2 = 18; Possible
B. x = 3; y = 6
a = 3s1 + 6 ; b = 6s2 + 21; 3s1 + 6 = 6s2 + 21; 3s1 - 6s2 = 15; s1- 2s2 = 5; Possible
C. x = 7; y = 9
a = 7s1 + 28 ; b = 9s2 + 45; 7s1 + 28 = 9s2 + 45; 7s1 - 9s2 = 17; Possible
D. x = 10; y = 4
a = 10s1 + 55 ; b = 5s2 + 10; 10s1 + 55 = 4s2 + 10; 10s1 - 4s2 = -45; Impossible since 10s1 and 4s2 are even but 45 is odd
E. x = 10; y = 7
a = 10s1 + 55 ; b = 7s2 + 28; 10s1 + 55 = 7s2 + 28; 10s1 - 7s2 = -27; Possible


IMO D

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