Which of the following could be the sum of 12 consecutive integers?

Answer - 78 must be divisible by 12

The sum of 12 consecutive integers can be expressed using the following equation:
x + x+1 + x+2 + ..... + x+12 ----> 12x + the sum of the integers between 1 and 12 inclusive, 12 * (L + F)/2 = 78

12x + 78 = sum of 12 consecutive integers, at this point should be crystal clear that a number could be the sum of 12 consecutive integers if and only if subtracting 78 from this number it results divisible by 12 -----> B is obviously the answer (198-78= 120 divisble by 12)

Hope this help!

Trial and error is the best way I guess?

12 consecutive integers : 12x + 66 <-- Equate this with each choice to see which one gives an integer for x.

B.

Sum of consecutive integers can be written as: (mean) x (number of integers) = mean x 12
Therefore the answer choice must be divisible by both 12 and the mean


For a number to be divisible by 12 it must be divisible by 2 and 3 (prime factorize 12):

divisible by 2: Last number must be divisible by 2
= All answer choices are divisible by 2

divisible by 3: Sum of numbers must be divisible by 3
(a) 92: 9+2 = 11 NOT DIVISIBLE BY 3
(b) 198: 1+9+8 = 18 DIVISIBLE BY 3 (18/3 = 6)

IMHO answer choice (b) is correct

Solution:

If the first integer is x, then the second integer is x + 1, the third integer is x + 2, and so on. Therefore, the sum of the 12 consecutive integers, in terms of x, is:

x + (x + 1) + (x + 2) + … + (x + 11) = [x + (x + 11)]/2 * 12 = (2x + 11) * 6 = 12x + 66

Now, let’s check the answer choices starting from choice A:

12x + 66 = 92

12x = 26

x = 26/12 = 13/6 = 2 1/6

Since x has to be an integer, the sum of the 12 consecutive integers could not be 92.

Let’s check choice B:

12x + 66 = 198

12x = 132

x = 132/12 = 11

Since x is an integer, the sum of the 12 consecutive integers could be 198.

Alternate solution:

Notice that the sum of the 12 consecutive integers, in terms of x, is 12x + 66, which is a multiple of 6, i.e., a multiple of both 2 and 3. We see that all the answer choices are even; thus, they are all multiples of 2, and therefore, it hinges on which answer choice is also a multiple of 3. Recall that, if the sum of the digits of the number is a multiple of 3, then the number itself is a multiple of 3. Looking at the choices, we see that sums of the digits of the numbers are 11, 18, 13, 7, and 13, respectively, and only 18 is a multiple of 3. Therefore, 198 is the correct answer.

Answer: B

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.