2011/015) Let p be a prime number > 5. Show that p divides infinitely many of the numbers 1, 11, 111 ...

as p > 5 p and 10 are coprimes and so are p and 9

so if p devides x <=> p devides 9x

so p devides 9, 99, 999, so on

so 10^n mod p = 1

now as p and 10 are co-primes then

10^n(p-1) is 1 mod p(for n = 1 ... )

or 10^n(p-1) -1 is divsible by p

or (10^n(p-1)-1)/9 that is 1 repeated n(p -1) is dvisible by p

so there are many numbers divisible by p