Wednesday, April 22, 2015

Prove that $10152^8-10887^8+27195^8$ is divisible by $26460$.

The very first thing that we need to do is to factorize $26460$ as [MATH]\color{yellow}\bbox[5px,purple]{26460=2^2\cdot 3^3\cdot 5\cdot 7^2}[/MATH]. The next thing we have to accomplish is to show that $10152^8-10887^8+27195^8$ is divisible by $2^2\cdot 3^3\cdot 5\cdot 7^2$, either

1. all at once or
2. separately.

But, I am sure you will also work out the prime factors for the other three numbers

$10152=2^3\cdot 3^3\cdot 47$; $10887=3\cdot 19\cdot 191$; $27195=3\cdot 5\cdot 7^2\cdot 37$

and notice immediately that the each of the term in the given expression is not divisible by each of the numbers [MATH]\color{yellow}\bbox[5px,purple]{2^2,\,3^3,\,5,\,7^2}[/MATH] since e.g. $10152=2^3\cdot 3^3\cdot 47$ has not the factor of $5$ nor $7^2$.

So, we need to devise one good plan of attack, that is, we need to show, separately, that the given expression is divisible by each of the numbers [MATH]\color{yellow}\bbox[5px,purple]{2^2,\,3^3,\,5,\,7^2}[/MATH], so that the result will follow.

What does this mean exactly? You might wonder and begin to fret and feeling glum. Yes, that is all we tend to do, when something doesn't immediately obvious to us, and we get the sense that can't ultimately control the math problem that is thrown to us, it is then easy to opt for the coward way out, to give up, hehehe...

But, at this blog, we are the devout advocate for creating learning opportunity to shape and further instilling in every student the core values that they must possess to become critical thinker. Continue reading, and rest assure you will learn something useful for the day!

We will at first show the general principle to let you witness how we could prove an expression that contains the addition and subtraction operation is divisible by a product of some prime numbers, by proving the expression, separately, be divisible by that product of prime numbers.

In any mathematical problems, words won't do them justice, let's begin to work for the explanation mathematically:

If $A^8-B^8+C^8=(2^2\cdot 3^3\cdot 5)(C+D+E)$, it's no doubt that $A^8-B^8+C^8$ is divisible by $2^2\cdot 3^3\cdot 5$.

On the other hand, if $A^8-B^8+C^8=(7^2)(F+G+H)$, we can tell $A^8-B^8+C^8$ is also divisible by $7^2$.

So together, we could conclude that

$A^8-B^8+C^8$ is divisible by

1. $2^2\cdot 3^3\cdot 5$ and

2. $7^2$, that is

$A^8-B^8+C^8$ is divisible by $2^2\cdot 3^3\cdot 5\cdot 7^2$.

Okay, we have the general concept in mind, what we need to notice here is we need to group the differences of squares so we could further simplify the expression by applying all the short cut hints we already know, i.e. the difference of two squares where $x^2-y^2=(x+y)(x-y)$:

We could work on grouping $10152^8-10887^8$ first:

$10152^8-10887^8+27195^8$

$=(10152^8-10887^8)+27195^8$

$=((10152^4)^2-(10887^4)^2)+27195^8$

$=(10152^4+10887^4)(10152^4-10887^4)+27195^8$

$=(10152^4+10887^4)(10152^4-10887^4)+27195^8$

$=(10152^4+10887^4)(10152^2+10887^2)(10152^2-10887^2)+27195^8$

$=(10152^4+10887^4)(10152^2+10887^2)(10152+10887)(-735)+27195^8$

$=27195^8-(10152^4+10887^4)(10152^2+10887^2)(10152+10887)(735)$

$=(3\cdot 5\cdot 7^2\cdot 37)^8-(10152^4+10887^4)(10152^2+10887^2)(21039)(3\cdot 5\cdot 7^2)$

$=(3\cdot 5\cdot 7^2)((3\cdot 5\cdot 7^2)^6(37)^8-(10152^4+10887^4)(10152^2+10887^2)(21039))$

This partly proved that $10152^8-10887^8+27195^8$ is divisible by $3\cdot 5\cdot 7^2$.

Next we need to group $-10887^8+27195^8$ together:

$10152^8-10887^8+27195^8$

$=(2^3\cdot 3^3\cdot 47)^8+27195^8-10887^8$

$=(2^3\cdot 3^3\cdot 47)^8+((27195^4)^2-(10887^4)^4)$

$=(2^3\cdot 3^3\cdot 47)^8+(27195^4-10887^4)(27195^4-10887^4)$

$=(2^3\cdot 3^3\cdot 47)^8+(27195^4-10887^4)((27195^2)^2-(10887^2)^2)$

$=(2^3\cdot 3^3\cdot 47)^8+(27195^4-10887^4)(27195^2+10887^2)(27195^2-10887^2)$

$=(2^3\cdot 3^3\cdot 47)^8+(27195^4-10887^4)(27195^2+10887^2)(27195+10887)(27195-10887)$

$=(2^3\cdot 3^3\cdot 47)^8+(27195^4-10887^4)(27195^2+10887^2)(27195+10887)(16308)$

$=(2^3\cdot 3^3\cdot 47)^8+(27195^4-10887^4)(27195^2+10887^2)(27195+10887)(2^2\cdot 3^3\cdot 151)$

$=(2^2\cdot 3^2)((2^2\cdot 3^2)^7(2\cdot 3\cdot 47)^8+(27195^4-10887^4)(27195^2+10887^2)(27195+10887)(151))$

We have proved that $10152^8-10887^8+27195^8$ is also divisible by $2^2\cdot 3^2$ as well.

So we can conclude that $10152^8-10887^8+27195^8$ is divisible by $3\cdot 5\cdot 7^2$ and $2^2\cdot 3^2$, which merged into $2^2\cdot 3^3\cdot 5\cdot 7^2=26460$ and we are hence done!