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Without using a calculator, find the largest prime divisor of 5^12 - 3^12.

 Dec 31, 2020
 #1
avatar+94 
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Hello Guest!

 

I tried to solve the problem, but ended not being able to. 

However, here's what I got. 

 

a^2 - b^2 = (a+b)(a-b)

a^2 + b^2 = (a + b)^2 – 2ab

 

Using a^2 - b^2 = (a+b)(a-b), we can simplify the original equation to: 

(5^6 - 3^6)(5^6 + 3^6)

Then, we can use that same equation to simplify the "(5^6 - 3^6)" portion:

(5^3 + 3^3)(5^3 - 3^3)(5^6 + 3^6)  

5^3 and 3^3 are calculatable without a calculator 5^3 + 3^3 = 152, 5^3 - 3^3 = 98. Plugging those in:

(152)(98)(5^6 + 3^6)

Now for the "(5^6 + 3^6)" part, we'll use the formula (a + b)^2 – 2ab. 

(5^3+3^3)^2 – 2*(5^3*3^3)

 

(5^3+3^3)^2 – 2*(45^3)

(152)^2 – 2*(15^3)

(2^6 * 19^2) - (2 * 3^3 * 5^3)

 

And then... I got stuck. :(((((

 

However, I hope that someone else on the forum will be able to help you. :))))

 

=^._.^=

 Dec 31, 2020
 #2
avatar+428 
0

Adding on to catmg's post:

 

If you know 2048, you know 2^6 = 64.

 

And also 19^2 is memorizable, it's 361.

 

But note that 64 = 8^2, so it's really just (8 * 19)^2, which is 152^2.

Then 2 * 3^3 * 5^3 = 125 * 2 * 27, or 250 * 27, and you can use the x25 rule to get 6750.

152^2 = 152 * 152  = 04 _ (15 * 2 + 15 * 2) + 15^2 = 04 _ 60 + 225 = 23104, by LOIFing.

 

Thus, easy subtraction to get 16354. Divide it by 2 (long division) and reach 8177, which is easily divisible by 13, = 629, which is easily divisible by 17, 37, which is a prime.

 

And as none of the other factors 152 or 98 are divisible by 37, the answer is 37!!!

 Dec 31, 2020

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