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Operations on integers

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Operations on integers

by jmacym » Thu Sep 29, 2016 10:36 am
I have the 2017 GMAT prep book and number 98 in "Problem Solving" has me confused.

The sum of all the integers k such that -26 < k < 24 is .....

A 0
B -2
C -25
D -49
E -51

There is an answer explanation of why it is D, but I don't follow how they got a value for K and therefore answer D.
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by regor60 » Thu Sep 29, 2016 11:14 am
jmacym wrote:I have the 2017 GMAT prep book and number 98 in "Problem Solving" has me confused.

The sum of all the integers k such that -26 < k < 24 is .....

A 0
B -2
C -25
D -49
E -51

There is an answer explanation of why it is D, but I don't follow how they got a value for K and therefore answer D.
K is just a variable representing any integer that satisfies the relation. The problem asks you to sum all of these "K"s

There are 25 negative integers that satisfy: -1,-2,...-25. 0 also satisfies, as do the integers 1,2,3,...23

Now having to add all of these, notice that conveniently the positive and negative numbers all sum to 0, except that there are no corresponding positive numbers to offset -24 and -25.

So, the net sum is -49
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by Brent@GMATPrepNow » Thu Sep 29, 2016 11:31 am
jmacym wrote: The sum of all the integers k such that -26 < k < 24 is .....

A 0
B -2
C -25
D -49
E -51
(-25) + (-24) + (-23) + (-22) + (-21) + . . . . .+ 21 + 22 + 23

Notice that the blue values all cancel out, because for each positive value, there's a negative value with the same magnitude (e.g., -23 + 23 = 0, and -22 + 22 = 0, etc)

So, we're left with (-25) + (-24), which equals -49
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by Matt@VeritasPrep » Thu Sep 29, 2016 6:32 pm
I think what you're asking for is a translation of the math, so let's just do that:

k is any integer that satisfies -26 < k < 24. So k could be 10, since -26 < 10 < 24, and k could also be -2, since -26 < -2 < 24.

"all the integers k" means that we want EVERY value of k that could work. k could be -25, -24, -23, -22, ..., 21, 22, 23, so we want the sum of ALL of these, or -25 + -24 + -23 + -22 ... + 22 + 23.

Notice all that all the negatives and positives will cancel out, EXCEPT for -25 and -24! So that's the sum, -25 + -24, or -49.
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by Matt@VeritasPrep » Thu Sep 29, 2016 6:34 pm
It might be easiest to see if we color code it:

-25 + -24 + -23 + -22 + -21 + ... + 21 + 22 + 23

All those pairs with matching colors will sum to 0, so they vanish, leaving only -25 + -24.
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