Problem Solving

BTGModeratorVI wrote: ↑

Wed Oct 07, 2020 7:10 am

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

Answer: E
Source: Princeton Review

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

BTGModeratorVI wrote: ↑

Wed Oct 07, 2020 7:10 am

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

Answer: E

Solution:

If a number can be written as both the sum of 5 consecutive odd integers and 7 consecutive odd integers, then it’s a multiple of both 5 and 7. This eliminates choice A, 49 (since it’s only a multiple of 7) and choice D, 215 (since it’s only a multiple of 5). So we are left with 70, 140 and 525 to consider.

Not only must the sum be both a multiple of 5 and 7, but the sum also has to be odd (since the sum of five odd numbers must also be odd). This eliminates answer choices B and C and we are left with E.

Answer: E