A list contains 7 positive integers

Question

A list contains 7 positive integers. What is the value of the smallest integer in the set?

(1) The highest term in the set is 37
(2) When arranged in increasing order, the difference between any two consecutive terms is  at least  6

gmatophobia · Answer

Given  
  The list has 7 integers 
 All the integers are positive

Question   - What is the value of the smallest integer in the set?

Statement 1

The highest term in the set is 37

Merely knowing the highest terms doesn't help. The smallest integer can be any value between 1 and 37, both inclusive.

The statement is insufficient and we can eliminate A and D

Statement 2

When arranged in increasing order, the difference between any two  consecutive  terms is  at least  6

Just like Statement1, Statement 2 doesn't provide much information. We can have infinite possibilities of the lowest term.

Eliminate B

Combined

The statements combined tells us that maximum value is 37 and the difference between two any two consecutive terms is  at least  6.

Hence, if we consider the difference to be exactly 6, the lowest term is 1. If we increase the difference between any two consecutive terms, the lowest term will not be positive.

Hence we have an unique answer.

Option C

SaquibHGMATWhiz · Answer

Solution:      
  Pre Analysis:  
  A list contains 7 positive integers 
 We are asked the smallest integer in the set

Statement 1:   The highest term in the set is 37
  This will not help us get the smallest term in the set 
 Thus,  statement 1 alone is not sufficient and we can eliminate options B, C and E

Statement 2:   When arranged in increasing order, the difference between any two consecutive terms is  at least  6
  We are given difference between any two consecutive terms = 6 
 From this we cannot say what the smallest term in the set is 
 Thus,  statement 2 alone is also not sufficient

Combining:   
  Smallest positive integer is 1. So, let us assume the smallest term in the list is 1 
  If the first term (smallest) is 1 and the seventh term (highest) is 37, then the difference between any two has to be 6  and not more than that  
 Thus, upon combining it is sufficient to say that the smallest term in the list is 1

Hence the right answer is    Option C

luisdicampo · Answer

Deconstructing the Question

Let the 7 positive integers in increasing order be   a_1<a_2<a_3<a_4<a_5<a_6<a_7  .

We need the value of the smallest integer, which is   a_1  .

Statement (1) gives the largest term.
Statement (2) gives the minimum gap between consecutive terms.

The key is to check whether these conditions force a unique value for   a_1  .

Step-by-step

Statement (1): The highest term in the set is   37  .

So   a_7=37  .

This alone is not enough. The smallest term could vary.

Statement (1):  Insufficient

Statement (2): The difference between any two consecutive terms is at least   6  .

So each gap satisfies

a_2-a_1\ge 6,\ a_3-a_2\ge 6,\ \dots,\ a_7-a_6\ge 6

But there is no fixed largest or smallest value, so the smallest term can vary.

Statement (2):  Insufficient

Now combine the statements.

We know   a_7=37  , and there are   6   gaps between the   7   numbers.

Since each gap is at least   6  , the total difference between largest and smallest is at least

6\cdot 6=36

So

a_7-a_1\ge 36

Substitute   a_7=37  :

37-a_1\ge 36

Thus

a_1\le 1

But all integers are positive, so

a_1\ge 1

Therefore,

a_1=1

Final Answer:  C

A list contains 7 positive integers

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A list contains 7 positive integers

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