LCT n pg Where p and g are distinct primes with p g 000 Show

LCT n= pg Where p and g are distinct primes, (with p, g 000). Show that p-1 times n-1 or g-1 times n-1

Solution

Let, p-1 |n-1

So, n-1=k(p-1)

pq-1=n(p-1)

pq-1=np-n

n-1=p(n-q)

n has distint odd primes as factors so n is not equal to 1 and n not equal to q

So, p|n-1 which is not possible as p|n

Hence a contradiction

So, p-1 does not divide n-1

Similarly q-1 does not divide n-1