Is there any other way to solve following problem , other than explained in CAT explanation ?:

The sum of n consecutive positive integers is 45. What is the value of n?

(1) n is even

(2) n < 9


Explanation given :

The possible values of n should be computed right away, to rephrase and simplify the question. Note that n consecutive positive integers that sum to 45 have a mean of 45/n, which is also the median of the set; therefore, the set must be arranged around 45/n. Also, any set of consecutive integers must have either an integer mean (if the number of integers is odd) or a mean that is an integer + 1/2 (if the number of integers is even). So, if we compute 45/n and see that it is neither an integer nor an integer +1/2, then we can eliminate this possibility right away.

Setting up a table that tracks not only the value of n but also the value of 45/n is useful.

n 45/n n positive consecutive integers summing to 45

1 45 45

2 22.5 22, 23

3 15 14, 15, 16

4 11.25 none

5 9 7, 8, 9, 10, 11

6 7.5 5, 6, 7, 8, 9, 10

7 6 3/7 none

8 5 5/8 none

9 5 1, 2, 3, 4, 5, 6, 7, 8, 9

10 4.5 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 -- but this doesn't work, because not all are positive integers

... ... impossible (the set will include negative integers, if an integer set can be found at all)



(1) INSUFFICIENT: If n is even, n could be either 2 or 6. Statement (1) is NOT sufficient.

Alternatively, to find these values algebraically, you can use the following procedure.
The sum of two consecutive integers can be represented as n + (n + 1) = 2n + 1
The sum of three consecutive integers = n + (n + 1) + (n + 2) = 3n + 3
The sum of four consecutive integers = 4n + 6
The sum of five consecutive integers = 5n + 10
The sum of six consecutive integers = 6n + 15

Since the expressions 2n + 1 and 6n + 15 can both yield 45 for integer values of n, 45 can be the sum of two or six consecutive integers.

(2) INSUFFICIENT: If n < 9, n could again take on either of the values 2 or 6 (or 3 or 5 according to the table or the expressions above)

(1) and (2) INSUFFICIENT: if we combine the two statements, n must be even and less than 9, so n could still be either of the values: 2 or 6.

The correct answer is E.

Of course there is. But remember you only have 75 minutes to complete the math section, so the advanced number theory you learned in grad school is probably overkill. :) Notice how the statements give very vague information about the number; this is usually a giveaway that the best way to solve the problem is to rephrase the question to identify possible values of n..

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Tim Sanders
Manhattan GMAT Instructor

Hi there,

The one thing I cant understand is how do you arrive at this conclusion with the algebraic approach:

"Since the expressions 2n + 1 and 6n + 15 can both yield 45 for integer values of n, 45 can be the sum of two or six consecutive integers"

Wouldnt "The sum of five consecutive integers = 5n + 10" also yield an integer for n when 45? 45-10=35/5=7=n

Im sure it doesnt work like that, but I cant figure out why from the explanation...

You're absolutely right. n could be 5, with numbers 7, 8, 9, 10, 11 summing to 45. However, we are only interested in even values for n for statement 1.

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Jamie Nelson
ManhattanGMAT Instructor

Sum of n consecutive numbers:
= (n/2)(2a+n-1) = 45 , where a is the first number of the series

now from statement 1
n is even,

so n could be 2,4,6,8...
for n=2 , the numbers could be 22,23
you can leave out n=4 since it shall contain 2 odd and 2 even numbers, the sum of which can't be 45(odd)
for n=6 , the numbers would be 5,6,7,8,9,10 (just put n=6 and evaluate for a)

So Statement 1 is insufficient since we can't conclude on a single value of n

From statement 2
n < 9
we already inferred from statement 1 that for both n=2 and n=6 the equation holds true. So we can't conclude on a value for n
so statement 2 is also insufficient

Together: n is even and n is < 9
Same logic as for statement 2 can be used.

Answer should be E

As always, correct atul.

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Jamie Nelson
ManhattanGMAT Instructor

Great explanation. exactly what I was looking for...

Great! Thanks a ton atul!

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Jamie Nelson
ManhattanGMAT Instructor