CAT Quantitative Aptitude Questions | CAT Algebra - Inequalities questions

CAT Questions | Algebra | Inequalities - Integer Solutions

The question is about maximum possible value. Sum of three distinct number is given and our task is to maximize their product. Inequalities are crucial to understand many topics that are tested in the CAT exam. Having a good foundation in this subject can help us tackle questions in Coordinate Geometry, Functions, and most importantly in Algebra. A range of CAT questions can be asked based on this simple concept.

Question 10: If a, b, c are distinct positive integers, what is the highest value a * b * c can take if a + b + c = 31?

  1. 1080
  2. 1200
  3. 1024
  4. 1056

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Explanatory Answer

Method of solving this CAT Question from Algebra - Inequalities : When 'a' + 'b' is given to be a constant, 'ab' has its maximum value when they are closest to each other, or if 'a' = 'b'. What happens when 'a' and 'b' are distinct?

The sum of three numbers is given; the product will be maximum if the numbers are equal.
\\frac{a + b + c}{3}\\) > ∛abc
So, if a + b + c is defined, abc will be maximum when all three terms are equal. In this instance, however, with a, b, c being distinct integers, they cannot all be equal.

So, we need to look at a, b, c to be as close to each other as possible.
a = 10, b =10, c = 11 is one possibility, but a, b, c have to be distinct. So, this can be ruled out.
The close options are,
a, b, c: 9, 10, 12; product = 1080
a, b, c: 8, 11, 12; product = 1056
Maximum product = 1080

The question is "what is the highest value a * b * c can take if a + b + c = 31?"

Hence the answer is "1080"

Choice A is the correct answer.


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