For any integers x and y, min(x, y) and max(x, y) denote the minimum a

Question

For any integers x and y, min(x, y) and max(x, y) denote the minimum and the maximum of x and y, respectively. For example, min(2, 1) = 1 and max(2,1) = 2. If a, b, c and d are distinct positive integers, is max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))) ?

(1) b, c and d are factors of a

(2) a – 2d = b + c

(1) b, c and d are factors of a
(2) a – 2d = b + c

chetan2u · Answer

max(a, max(b, min(c, d))) ... You will get MAX value out of the 4 only if a or b are max....
 max(d, max(a, min(b, c))) .... You will get the max value out of the four only if a and d are max..

So we can say max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))) ?
In two conditions..
1) if A is the max
2) if C is the max as it will get discarded in MIN and we will get the second largest from the remaining 3

Let's see the statements..
(1) b, c and d are factors of a.
Since all are different, we can say a is largest.
Meets our first criterion, is Ans will be a from both sides
Suff
(2) a-2d=b+C
a=b+c+2d
So a is largest..
Suff

D

rahul16singh28 · Answer

Statement I:

a is the maximum of b,c,d. So,  max(a, max(b, min(c, d)))  &  max(d, max(a, min(b, c)))  will always be a.

Statement II:

a = b + c + 2d.  As,  a, b, c & d  are distinct  POSITIVE  Integers, a will always be maximum.

max(a, max(b, min(c, d)))  &  max(d, max(a, min(b, c)))  will always be a.

Hence, D.

CareerGeek · Answer

(1) b, c and d are factors of a
--> b, c & d are less than are equal to a 
max(a, max(b, min(c, d))) = max(d, max(a, min(b, c)))
--> max(a, max(b, (c/d))) = max(d, max(a, (b/c)))
--> max(a, {b or (c/d)}) = max(d, {a or (b/c)})
--> a = a (Since, the maximum value of a,b,c,d is a) -  Sufficient

(2) a – 2d = b + c
--> a = 2d + b + c
Since, a, b, c & d are positive. Maximum = a
So, as per (1) -  Sufficient .

Option D

Pls Hit kudos if you like the solution

CrackverbalGMAT · Answer

Although this question looks like a difficult one, it’s actually not difficult when you plug in the right values. The fact that a, b, c and d are distinct positive integers actually makes life easier.

From statement I alone, since b, c and d are factors of a, we can conclude that b, c and d are smaller than a or utmost equal to a.

If b, c and d are smaller than a, then the max value in both the brackets will be a. We will be able to answer the question with a YES.

If all of b, c, d and a are equal, then, the value on both sides will come out to be a. Let’s take an example:

Let a = 4, b = 4, c = 2 and d = 1. Then,

max(4, max(4, min(2,1))) = max (1, max(4,min(4,2))), which on simplification, gives us,

max(4, max(4,1)) = max(1, max(4,2)) which in turn gives us,

max(4,4) = max(1,4).

Clearly, both sides are equal. If we take a=b=c=d=4, the two sides will again will be equal. This data is sufficient. Answer options can be A or D. Options B, C and E can be ruled out.

From statement II alone, we can say that a= b + c + 2d. Since all of the numbers are distinct positive integers, clearly a has to be the biggest. In this case, again, both the LHS and RHS will yield a and hence both sides will be equal. Hence, this data is also sufficient.

The correct answer option is D.

When there are problems on maximization and minimization, the idea is to consider two scenarios – one, when all the values are equal and hence maximum or minimum will be the same; two, when we have one value which is bigger than all others so that you can clearly distinguish between maximum and minimum.

Hope this helps!

vishumangal · Answer

ArvindCrackVerbal  : In the Solution provided by you, random values of a and b are taken same.
I guess as per the question stem a,b,c,d suppose to be distinct integers so the values of and b cannot be same. Although, It will not effect the final outcome, but just for the clarity

CrackverbalGMAT · Answer

That's right Vishnumangal. As we take bigger values for a (let's say a=6), a will have more factors and therefore, we will be able to take distinct values for a, b, c and d. For example, we can take a = 6, b = 3, c = 2 and d = 1 However, as you have seen, taking all the values as same (although not permitted by the question) is like trying out the boundary condition. It's like saying "Okay, the values at the extremities satisfy my constraints, so the values within the extremities should also hold true". In other words, "In the worst case scenario that all the values are equal also, 'a' turns out to be the maximum. When we consider 'a' to be the biggest of the lot, it HAS to be the maximum regardless of the values of b, c and d". Does this sound right? Please let me know. Thanks

MHIKER · Answer

What wrong will I do if I that as both conditions are giving specific values for all the variables so we can solve the given equation? And the answer will be.

Actually, I took four value such as

d=1
c=2
b=3, and 
a= 6

Then I solved the equation. Later on, I realized this is a DS and I can just say Yes and No. I am asking you to double-check my thinking. I regularly watch the webinar by CrackVerbalGMAT.

Hope you will understand my concern.

MHIKER · Answer

(1) As a, b, c and d are distinct positive integers so according to this condition, a will be the largest number. In the both part of max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))) "a'' will remain, i.e., a=a; Sufficient.

(2) a – 2d = b + c; a=b+c+2d, which means a is the largest value and In the both part of max(a, max(b, min(c, d))) = max(d, max(a, min(b, c))) "a'' will remain, i.e., a=a; Sufficient.

The answer is  D

MayankThapar · Answer

Hi, I have a doubt here. For statement 1, when we reach the final stage of max(a,d) - given that d is a factor of a, why can't d=a, since a number is also a factor of itself.

For any integers x and y, min(x, y) and max(x, y) denote the minimum a

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For any integers x and y, min(x, y) and max(x, y) denote the minimum a

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