How Can You Maximize the Product of Two Numbers with 100 Good Numbers?

[anemone]

Here is this week's POTW:

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A man chooses two positive integers $a$ and $b$. He then defines a positive integer $k$ to be good if a triangle with side lengths $\log a$, $\log b$ and $\log k$ exists. He finds that there are exactly $100$ good numbers. Find the maximum possible value of $ab$-----

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[anemone]

Congratulations to kaliprasad for his correct solution::)

Solution from kaliprasad:

Spoiler

Without loss of generality we can assume $a\ge b$.

As $\log\, a$, $\log\, b$ and $\log\, k$ form a triangle we have $\dfrac{a}{b}< k < ab$.

So maximum value of k is ab-1 and minimum $\lfloor\dfrac{a}{b}+1\rfloor$.

As there are 100 values of k we have:

$ab-1 - \lfloor\frac{a}{b}+1\rfloor = 99$

or $ab - \lfloor\frac{a}{b}\rfloor= 101$

For the same $a$, as $b$ increases $\frac{a}{b}$ decreases and so $ab$ decreases and we expect it to be larger when $b$ is in minimum.

$b\ge 2$ (as it is integer ) else $\log\, b$ is undefined.

Put $b = 2$ to get $a = 67$ and $ab= 134$ is maximum.

(Note $b= 5$ and $a = 21$ and $ab = 105$ is minimum and these are only 2 solutions for $ab$.)