If p is a prime an p is not divisible to b, prove that the arithmetic progression a, a+b, a+2b, a+3b,…… every pthe term is divisible by p....

If p is a prime an p is not divisible to b,
prove that the arithmetic progression a, a+b, a+2b, a+3b,…… every pthe term is divisible
by p.

Consider the sum(a+kb)(mod p). This is
equivalent to a(mod p) + (kb)(modp). Since p does not divide b, the sequence formed by
kb(mod p) as k goes from 1 to p forms a permutation, taking on every value from 1 to
p-1, with kb(mod p)=0 when k=p. Thus the sum (a+kb)(mod p) takes on every value from 0
to p-1 as k ranges from 1 to p. At the k where (a+kb)(mod p)=0, then p|(a+kb). Let this
be k'.

The permutation repeats, so the sum is divisible by
p every npk' where n is an integer. Thus p|(a+kb) every pth
term.