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The sum of 4 different odd integers is 64. What is the value

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The sum of 4 different odd integers is 64. What is the value

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The sum of 4 different odd integers is 64. What is the value of the greatest of these integers?

(1) The integers are consecutive odd numbers
(2) Of these integers, the greatest is 6 more than the least.

OA D

Source: Official Guide

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BTGmoderatorDC wrote:
The sum of 4 different odd integers is 64. What is the value of the greatest of these integers?

(1) The integers are consecutive odd numbers
(2) Of these integers, the greatest is 6 more than the least.
Source: Official Guide
$$\sum\nolimits_{4\,\,{\rm{different}}\,\,{\rm{odds}}} {\,\, = \,\,\,64\,\,\,\,\left( * \right)} $$
$$? = \,\,{\rm{max}}\,\,{\rm{among}}\,\,{\rm{them}}$$

$$\left( 1 \right)\,\,\,{\rm{consecutive}}\,\,{\rm{and}}\,\,{\rm{sum}}\,\,64\,\,\,\left( {{\rm{from}}\,\,\left( * \right)} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\rm{they}}\,\,{\rm{are}}\,\,{\rm{unique}}!\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\rm{SUFF}}.\,\,\,\,$$
\[\left( 2 \right)\,\,\,{\text{must}}\,\,{\text{be}}\,\,{\text{consecutive}}\,\,\,\left[ {\,\,\underline {2M - 3} \,\,,\,\,2M - 1\,\,,\,\,2M + 1\,\,,\,\,\underline {2M + 3} \,\,} \right]\,\,\,\,\,\, \Rightarrow \,\,\,\,\left( 1 \right)\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\text{SUFF}}.\,\]


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

Regards,
Fabio.

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Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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BTGmoderatorDC wrote:
The sum of 4 different odd integers is 64. What is the value of the greatest of these integers?

(1) The integers are consecutive odd numbers
(2) Of these integers, the greatest is 6 more than the least.
Statement 1:
Since the numbers are EVENLY SPACED, median = sum/count:
64/4 = 16.
Since the median = 16, the 4 odd integers must be as follows:
13, 15, 17, 19.
Thus, the greatest of the 4 odd integers = 19.
SUFFICENT.

Statement 2:
The 4 odd integers yielded by Statement 1 -- 13, 15, 17, 19 -- satisfy the condition that greatest - least = 6.
If we decrease the least and the greatest to 11 and 17, the middle 2 odd integers will be 13 and 15, with the result that the sum of the 4 integers = 11+13+15+17 = 56.
If we increase the least and the greatest to 15 and 21, the middle 2 odd integers will be 17 and 19 , with the result that the sum of the 4 integers = 15+17+19+21 = 72.
Implication:
Only the 4 odd integers yielded by Statement 1 -- 13, 15, 17, 19 -- will satisfy the conditions that the sum = 64 and greatest - least = 6.
Thus, the greatest of the 4 integers = 19.
SUFFICIENT.

The correct answer is D.

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