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a + b = \phi (a) + \phi (b) + gcd (a, b)

Source: 2020 Dutch IMO TST 2.3

November 22, 2020
number theoryrelatively prime

Problem Statement

Find all pairs (a,b)(a, b) of positive integers for which a+b=ϕ(a)+ϕ(b)+gcd(a,b)a + b = \phi (a) + \phi (b) + gcd (a, b).
Here ϕ(n) \phi (n) is the number of numbers kk from {1,2,...,n}\{1, 2,. . . , n\} with gcd(n,k)=1gcd (n, k) = 1.
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