If n is the product of the least and the greatest of 6 consecutive int

(1) The greatest of the 6 consecutive integers is 20.
--> Set of 6 integers = {15, 16, 17, 18, 19, 20}
--> Product, n = 15*20 = 300 --> Sufficient

(2) The average (arithmetic mean) of the 6 consecutive integers is 17.5.
--> (a + (a+1) + . . . . + (a+5))/6 = 17.5
--> 6a + 15 = 105
--> 6a = 90
--> a = 15
--> Product, n = 15*20 = 300 --> Sufficient

IMO Option D

Given: n = the product of the least and the greatest of 6 consecutive integers

Target question: What is the value of n?

Statement 1: The greatest of the 6 consecutive integers is 20.
Since the 6 integers are consecutive, we can work backwards to find the other 5 integers
They are {15, 16, 17, 18, 19, 20}
So, the product of the least and the greatest integers = (15)(20) = 300
The answer to the target question is n = 300
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The average (arithmetic mean) of the 6 consecutive integers is 17.5
There are several different ways to approach the statement.

APPROACH #1
The fastest approach is to recognize that for every possible set of 6 consecutive integers, we get a different average.
For example, the average of {2, 3, 4, 5, 6, 7} is 4.5, the average of {3, 4, 5, 6, 7, 8} is 5.5, the average of {4, 5, 6, 7, 8, 9} is 6.5, etc
So, if the average of of the 6 integers is 17.5, there must be EXACTLY ONE set of six consecutive integers that satisfies this information.
So, even if we were to just keep randomly checking various sets of 6 integers until we find the one set that has an average of 17.5, then we'd be able to determine the value of n.
Since we COULD use the information from statement 2 to answer the target question with certainty, statement 2 is SUFFICIENT

APPROACH #2
Another approach to handle statement 2 is as follows:
let x = the first (smallest) number in the set
So, x+1 = the next number in the set
x+2 = the next number in the set
x+3 = the next number in the set
x+4 = the next number in the set
x+5 = the last (greatest) number in the set

If the average = 17.5, we can write: \(\frac{x + (x+1)+ (x+2)+ (x+3)+ (x+4)+ (x+5)}{6 = 17.5}\)

Multiply both sides of the equation by 6 to get: \(x + (x+1)+ (x+2)+ (x+3)+ (x+4)+ (x+5) = 105\)
Simplify the left side to get: \(6x + 15= 105\)
Subtract 15 from both sides: \(6x= 90\)
Divide both sides by 6 to get: \(x = 15\)

Since x represents the first (smallest) value in the set, the 6 integers are {15, 16, 17, 18, 19, 20}
So, the product of the least and the greatest integers = (15)(20) = 300
The answer to the target question is n = 300
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent

Well done Dillesh!

The deduction from the first statement is perfect.

For the 2nd statement, however, we can simply do as follows:
In a set of terms having constant common difference (i.e. an AP - Arithmetic progression), the mean is simply the average of the first and the last terms
=> Sum of the 1st and the 6th number = 17.5 x 2 = 35
Since the difference between the 1st and the 6th numbers is 5:
# the 1st number = (35 - 5)/2 = 15
# the last number = (35 + 5)/2 = 20

=> product = 300

Answer D
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