A set of five distinct positive integers has a median of 3 and a range

Question

A set of five distinct positive integers has a median of 3 and a range of 12. What is the mean of the set of integers?

(1) The product of the integers in the set is a multiple of 14.
(2) The sum of the integers in the set is a multiple of 13.

Source: Advanced Quant Manhattan Prep 2020

AndrewN · Accepted Answer

The key to this question lies in processing the vocabulary of the problem and question stem. Here, we have  five  distinct  positive integers , so we know we are dealing with whole numbers (or counting numbers). We can create five slots to conceptualize our numbers:

___ ___ ___ ___ ___

If the  median  is 3, the middle number  must  be 3:

___ ___ 3 ___ ___

Furthermore, before we even jump across the comma, we know that the two integers to the left of 3  must  be 1 and 2, since they have to be distinct positive integers. (Remember, 0 is neither a positive nor negative number, the only one of its kind.)

1 2 3 ___ ___

Given a range of 12, we can quickly deduce that the number in the last slot  must  be 13, since 13 - 1 = 12.

1 2 3 ___ 13

All we need now to figure out the  mean  is the number that fits into the remaining slot. Since the two statements look about the same, I would just start with the first one. For the  product  to be a  multiple of 14 , we can take a look at the prime factorization of 14 for clues: 2 * 7. We know that 14 itself cannot be the missing number, since that would exceed 13 in value and throw off our range. Because the other numbers cannot be multiplied in any combination to yield 14, we know the number that fits into the blank  must  be 7.  Statement (1) is SUFFICIENT.  (With all the numbers in hand, we do not actually need to calculate the mean.) The answer has to be either (A) or (D).

At this point, we can use the test design against itself to assess the second statement. That is, the two statements  always  provide information that presents a consistent picture, even if that picture is not clear enough for you to be able to answer the question. You have to be careful not to carry information over from one statement to another, but at the same time, only 7 fits from statement (1), so we can think in the back of our minds before we process statement (2),  Does this statement reveal that the only number that fits is 7?  If the  sum  of the integers is a  multiple of 13 , then we would need the sum to be 13, 26, 39, and so on. Since 1 + 2 + 3 + 13 = 19, and again, we know that the fourth slot cannot exceed 13, we can only target 26 as the sum in question, and 19 + 7 = 26. The missing integer  must  be 7, so  statement (2) is SUFFICIENT .

With either statement sufficient on its own, we know that   the answer must be (D)  . I hope that the above proves helpful to anyone caught up in how to approach such a problem. Good luck with your studies.

- Andrew

gurmukh · Answer

From the main statement five numbers are 1,2 ,3, a, 13, because they are positive so only can start with 1 and they are distinct so first two values are 1 and 2 , also range is 12 so fifth values is 13. Now we have to find fourth value.
From first statement
The product is multiple of 14 and their is still no 7 in it so fourth number will be 7 as subsequent multiple of 7 cannot be taken because 13 is the highest number. Sufficient
From second statement
Sum is multiple of 13
Current sum 19+a
a can take value 7, 20 etc as they will make sum a multiple of 13 but we can't take value more than 13 so 7 is the only possible value we can take. Sufficient
Answer is option D

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Kinshook · Answer

Given: A set of five distinct positive integers has a median of 3 and a range of 12.
Asked: What is the mean of the set of integers?

(1) The product of the integers in the set is a multiple of 14.
14 = 2*7
Set of numbers = {1,2,3,7,13}
Mean = 26/5 = 5.2
SUFFICIENT

(2) The sum of the integers in the set is a multiple of 13.
Set of numbers = {1,2,3,7,13}
Mean = 26/5 = 5.2
SUFFICIENT

IMO D

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A set of five distinct positive integers has a median of 3 and a range

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