Tuesday, May 10, 2016

Compare which of the following is bigger: [MATH]1016^{11}\cdot 3016^{31}[/MATH] versus [MATH]2016^{42}[/MATH]

Compare which of the following is bigger:

[MATH]1016^{11}\cdot 3016^{31}[/MATH] versus [MATH]2016^{42}[/MATH]

My solution:

I know there are multiple approaches by which we can use to tackle the problem at hand. But if it were up to me, I would first look for the prime factorization for each number involved in the problem, and I found:

[MATH]1016=2^3\cdot 127[/MATH]

[MATH]2016=2^5\cdot 3^2\cdot 7[/MATH]

[MATH]3016=2^3\cdot 13\cdot 29[/MATH]

With the help of Wolfram Alpha, I've found

[MATH]1016^{11}\cdot 3016^{31}\gt 2016^{42}[/MATH]

The effort remains now to first assume [MATH]1016^{11}\cdot 3016^{31}\gt 2016^{42}[/MATH] is true and proved it so.

That is,

[MATH](2^3\cdot 127)^{11}\cdot (2^3\cdot 13\cdot 29)^{31}\gt (2^5\cdot 3^2\cdot 7)^{42}[/MATH]

Simplify the above we get:

[MATH]2^{33}\cdot 127^{11}\cdot 2^{93}\cdot 13^{31}\cdot 29^{31}\gt 2^{210}\cdot 3^{84}\cdot 7^{42}[/MATH]

[MATH]127^{11}\cdot 13^{31}\cdot 29^{31}\gt 2^{210-33-93}\cdot 3^{84}\cdot 7^{42}[/MATH]

[MATH]127^{11}\cdot 13^{31}\cdot 29^{31}\gt 2^{84}\cdot 3^{84}\cdot 7^{42}[/MATH]

[MATH]\color{yellow}\bbox[5px,purple]{127^{11}\cdot 377^{31}\gt 6^{84}\cdot 7^{42}}[/MATH]*

Here comes the difficult part of the process, we have to use the calculator to do a series of checking to see if we can get a simple relation between $377$ and $7^a$ for some positive integer $a$ such that $377\gt 7^a$.

Observe that:

$7^2=49,\,7^3=343\implies 373\gt 7^3$

Therefore $373^{14}\gt (7^3)^{14}$ which is $373^{14}\gt 7^{42}$

We have made some good progress, as we have kicked out the $7^{42}$ in the RHS of the equation * and also reduced the power of $377$ from $31$ down to $31-14=17$.

At this point, if we can prove

[MATH]\color{yellow}\bbox[5px,green]{127^{11}\cdot 377^{17}\gt 6^{84}}[/MATH]

Then we can conclude [MATH]1016^{11}\cdot 3016^{31}\gt 2016^{42}[/MATH] is correct.

Again, we have to find a relation between $127\cdot 377$ and $6^b$ for some positive integer $b$ such that $127\cdot 377\gt 6^b$.

Observe that

$127\cdot 377=47879\gt 6^6=46656$

Taking 14th power on both sides of the inequality we get:

$(127\cdot 377)^{14}\gt (6^6)^{14}$

[MATH]\color{black}\bbox[5px,orange]{127^{14}\cdot 377^{14}\gt 6^{84}}[/MATH]

It's obvious that

[MATH]\color{yellow}\bbox[5px,blue]{127^{11}\cdot 377^{17}\gt 127^{14}\cdot 377^{14} }[/MATH] is true, therefore, we've proved that

[MATH]1016^{11}\cdot 3016^{31}[/MATH] is bigger than [MATH]2016^{42}[/MATH].

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