Data Sufficiency

he average (arithmetic mean) of 5 distinct, single digit integers is 5. If two of the integers are discarded, the new average is 4. What is the greatest of these integers?

(1) Exactly 3 of the integers are consecutive primes.

(2) The least integer is 3.


Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.


Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.


BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.


EACH statement ALONE is sufficient.


Statements (1) and (2) TOGETHER are NOT sufficient.

The average of 5 distinct single digit integers is 5. If two of the integers are discarded, the new average is 4. What is the largest of the 5 integers?

(1) Exactly 3 of the integers are consecutive primes.

(2) The smallest integer is 3

Since the average of the 5 integers is 5, the sum of all 5 integers = 5*5 = 25.
After 2 integers are discarded, the average of the remaining 3 integers is 4, implying that the sum of the remaining 3 integers = 3*4 = 12.
Thus, the sum of the 2 discarded integers = 25-12 = 13.
Thus, the correct set of 5 integers must exhibit the following characteristics:
The integers are distinct and between 0 and 9, inclusive.
The sum of all 5 integers is 25.
The sum of 3 of the integers is 12.
The sum of the other 2 integers is 13.

Statement 1: Exactly 3 of the integers are consecutive primes.
Case 1: 2,3,5
Thus, the options for the remaining 2 integers are 0,1,4,6,8,9.
Since 2+3+5 = 10, the sum of the remaining 2 integers = 25-10 = 15.
Only one combination works: 6+9.
Thus, the 5 integers would be 2,3,5,6,9.
Not viable: this list does not include two integers whose sum is 13.

Case 2: 3,5,7
Thus, the options for the remaining 2 integers are 0,1,4,6,8,9.
Since 3+5+7 = 15, the sum of the remaining 2 integers = 25-15 = 10.
Only two combinations work: 1+9 and 4+6.
Thus, the 5 integers are either 1,3,5,7,9 or 3,4,5,6,7.
Only the second option includes two integers with a sum of 13 (6+7=13).
Thus, the 5 integers are 3,4,5,6,7.
SUFFICIENT.

Statement 2: The smallest integer is 3.
We know from statement 1 that 3,4,5,6,7 works.
For the smallest integer to be 3, the sum of the remaining 4 integers must be 25-3 = 22.
Given that these remaining 4 integers must each be greater than 3, there are no options aside from 4+5+6+7 = 22.
Thus, the 5 integers are 3,4,5,6,7.
SUFFICIENT.

The correct answer is D.