Loading [MathJax]/extensions/TeX/mathchoice.js

Wednesday, February 14, 2024

2024/013) Prove that 123123 | 2^{60}-1

To show that 123123 is a factor of 2^{60}-1 we need to show that each prime factor of 123123 to the highest power is a factor

Now let us factor 123123

123123= 123 * 1001 = 3 * 41 * 7 * 11 * 13 

As each prime occurs once we need to prove each of 3,7,11,13,41 divides  2^{60}-1

We know x^y-1 | x^{my}-1 for any m

We have 2^2-1 |  2^{60}-1  and 3| 2^{2} -1 using FLT(Fermat's Little Theorem)  so 3| 2{60}^-1

2^6-1 |  2^{60}-1  and 7| 2^{6} -1 using FLT(Fermat's Little Theorem)  so 7| 2{60}^-1

2^{10}-1 |  2^{60}-1  and 11| 2^{10} -1 using FLT(Fermat's Little Theorem)  so 11| 2{60}^-1

 2^{12}-1 |  2^{60}-1  and 13| 2^{12} -1 using FLT(Fermat's Little Theorem)  so 13| 2{60}^-1

Now we need to prove for 41

using FLT we know  41| 2^{40} -1

So let us find GCD(2^{(60}-1,2^{40}-1)

GCD(2^{(60}-1,2^{40}-1)= GCD((2^{(60}-1) - (2^{40-1},2^{40}-1)

=GCD((2^{60} - 2^{40}), 2^{40}-1)

=GCD((2^{40}( 2^{20}-1), 2^{40}-1)

 =GCD((2^{20}-1), 2^{40}-1) as second number is odd so removing even factors of 1st number

= 2^{20}-1 as this is a factor of  2^{40}-1

now = 2^{20}-1)= (2^{10}-1)(2^{10}+1)

=  1023 * 1025 as 41 | 1025 so 41 | 2^{60}-1

as each of 3,7,11,13,41 divides  2^{60}-1 so 123123 is a factor

No comments: