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
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