Problem solving: Difficulty level: 700, Source: Princeton

aishwaryav12 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

For any evenly spaced set:
sum = (number)(median)

We can PLUG IN THE ANSWERS, which represent the sum.
Since the number of integers can be 5 or 7, the sum must be a multiple of both 5 and 7.
Eliminate A, which is not divisible by 5.
Eliminate D, which is not divisible by 7.

The equation in blue can be rephrased as follows:
median = sum/number
Since all of the integers must be odd, the correct answer must yield an odd value for the median.

If B is the sum of 7 consecutive odd integers, we get:
median = sum/number = 70/7 = 10.
Since the yielded median is not odd, eliminate B.
If D is the sum of 7 consecutive odd integers, we get:
median = sum/number = 140/7 = 20.
Since the yielded median is not odd, eliminate D.

The correct answer is E.

aishwaryav12 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

$$?\,\,\,:\,\,\,\,\sum\nolimits_5 {\,{\rm{consecutive}}\,\,{\rm{odd}}\,\,{\rm{integers}}\,\,\,\, = \,} \,\,\,\,\sum\nolimits_7 {\,\,{\rm{consecutive}}\,\,{\rm{odd}}\,\,{\rm{integers}}} $$
> 5 consecutive odd integers are members of a finite arithmetic sequence, hence the middle term (one of them) is an odd integer and equal to their average.

(A) average 49/5 is not an integer, hence cannot be the middle term. Refuted.
(B) average 70/5 = 14 is not odd, hence cannot be the middle term. Refuted.
(C) average 140/5 = 28 is not odd, hence cannot be the middle term. Refuted.

> 7 consecutive odd integers are members of a finite arithmetic sequence, hence the middle term (one of them) is an odd integer and equal to their average.

(D) average 215/7 is not a multiple of 7 (210 is), hence cannot be the middle term. Refuted.

(E) is the answer by exclusion.

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.