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x is the sum of y consecutive integers. w is the sum of z

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by guerrero » Sat Jul 27, 2013 1:27 pm
x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT

1. x = w
2. x > w
3. x/y is an integer
4. w/z is an integer
5. x/z is an integer

OA 3
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by Matt@VeritasPrep » Sat Jul 27, 2013 2:10 pm
Let's pick numbers to illustrate a few of these.

1:: Say x = 0 + 1 and w = 1. x = w.
2:: Say x = 1 + 2 and w = 1. x > w.
4:: Say z = 1. Then w/z = an integer.
5:: Say z = 1. Then x/z = an integer.

So 1, 2, 4, and 5 are possible, and you're done!
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by Matt@VeritasPrep » Sat Jul 27, 2013 2:43 pm
I got a message asking me to explain why (3) is impossible, so here goes! Let me preface this explanation, however, by saying that I wouldn't waste time proving this on the exam - once you've proved that 1, 2, 4, and 5 are possible, 3 must be the answer.

If x is the sum of 2z consecutive integers, we could write x as an arithmetic progression:

x = n + (n + 1) + (n + 2) + ... + (n + 2z-1)
x = 2zn + (sum of integers from 1 to 2z-1)
x = 2zn + ((2z-1)(2z))/2
x = 2zn + (2z-1)(z)
x = 2zn + 2z^2 - z

So x/y = x/(2z) = (2zn + 2z^2 - z)/(2z) = n + z - 1/2

Since n and z are integers,

x/y = (an integer) - 1/2,

so x/y is not an integer.
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