The document discusses finding the maximum product of three distinct positive integers whose sum is 31. It explains that the maximum product occurs when the numbers are as close to each other as possible without being equal. The closest options that satisfy the distinct integer criteria are 9, 10, 12, which have a product of 1080. Therefore, the highest value the product a × b × c can take is 1080.

1. Inequalities

2. Inequalities
If a, b, c are distinct positive integers, what is the highest value
a × b × c can take if a + b + c = 31?
(a) 1080 (b) 1200
(c) 1024 (d) 1056

3. Inequalities
The sum of three numbers is given; the product will be maximum if
the numbers are equal.
a + b + c
3
>
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.
If a, b, c are distinct positive integers, what is the highest value
a × b × c can take if a + b + c = 31?

4. Inequalities
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
Answer choice (a)
If a, b, c are distinct positive integers, what is the highest value
a × b × c can take if a + b + c = 31?

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