When we studied prime numbers we looked at the Euler phi-function  ∅(n) = the number of integers between 1 and n with no factors in common with n            = n – (the number of integers between 1 and n with a factor in common with n) Look at the special case where n = pq, with p and q being different primes. Let A = {the integers between 1 and pq which are divisible by p}                 and argue that |A| = q Let B = {the integers between 1 and pq which are divisible by q}                 and argue that |B| = p Why is |A ∩ B|  = 1? Conclude that  |A U B|  =p + q -1  and that ∅(pq) = pq  - (p + q -1)  = (p-1)(q-1)

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When we studied prime numbers we looked at the Euler phi-function

 ∅(n) = the number of integers between 1 and n with no factors in common with n
           = n – (the number of integers between 1 and n with a factor in common with n)

Look at the special case where n = pq, with p and q being different primes.
Let A = {the integers between 1 and pq which are divisible by p}
                and argue that |A| = q
Let B = {the integers between 1 and pq which are divisible by q}
                and argue that |B| = p
Why is |A ∩ B|  = 1?
Conclude that  |A U B|  =p + q -1 
and that ∅(pq) = pq  - (p + q -1)  = (p-1)(q-1)

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