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.  Conclude that if b is an odd integer, then every...

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.

 

Conclude that if b is an odd integer, then every other term in the indicated progression is even.

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embizze | High School Teacher | (Level 1) Educator Emeritus

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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. 

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