2014/018) Find the unit digit of LCM of 3^2002 - 1 and 3^2002 + 1

As the difference of the 2 are 2 and both are even so the gcd = 2.

So we need to find the unit digit of ( 3^2002 – 1)*( 3^2002 + 1)/2

Or we need  (( 3^2002 – 1)*( 3^2002 + 1) mod 20) /2

So we need to find x mod 20 where x is ( 3^2002 – 1)*( 3^2002 + 1)

Let us find x mod 4 and x mod 5 as they are co primes

Now of ( 3^2002 – 1)*( 3^2002 + 1) as both are even mod 4 is odd

We have 3^2004-1 mod 4 = 0

As 3^2 = -1 mod 5 so 3^2004 = 1 mod 5 and 3^2004 – 1 = 0 mod 5

S0 3^ 2004 -1 = 0 mod 20 and hence it is multiple of 20 so (3^2004-1)/2 = 0 mod 10 so unit digit is zero