Carcass wrote:
Which of the following integers can be written as both the sum of 5 consecutive odd integers and 7 consecutive odd integers?
A. 49
B. 70
C. 140
D. 215
E. 525
Kudos for the right answer and explanation
If we have consecutive odd integers, and each value is 2 greater than the value before it
So for example, if we let x = the smallest odd integer in the set of five consecutive odd integers then...
The sum of the 5 integers = x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = 5x + 20 = 5(x + 4)
NOTE: since x is ODD, the sum of the 5 integers = 5(ODD + 4) = 5(some odd integer)
In other words, the sum must be an ODD MULTIPLE OF 5 (e.g., 5, 15, 25, 35, 45, etc)
Check the answer choices....
We can ELIMINATE A, B and C since they are NOT odd multiples of 5
Likewise, if we let k = the smallest odd integer in the set of seven consecutive odd integers then...
The sum of the 7 integers = k + (k + 2) + (k + 4) + (k + 6) + (k + 8) + (k + 10) + (k + 12) = 7k + 42 = 7(k + 6)
NOTE: since k is ODD, the sum of the 7 integers = 7(ODD + 6) = 7(some odd integer)
In other words, the sum must be an ODD MULTIPLE OF 7 (e.g., 7, 21, 35, 49, ... etc)
Check the remaining answer choices....
We can ELIMINATE D since it's not even a multiples of 7
By the process of elimination, the correct answer is E
Cheers,
Brent
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Brent Hanneson - founder of Greenlight Test Prep