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∑i=1mϕ(i∗n)=ϕ(p)∗∑i=1m[gcd(i,p)==1]ϕ(i∗n/p)+∑i=1m[p|i]ϕ(i∗n)=ϕ(p)∗∑i=1m[gcd(i,p)==1]ϕ(i∗n/p)+∑i=1m[p|i](p∗ϕ(i/p∗n))=ϕ(p)∗∑i=1m[gcd(i,p)==1]ϕ(i∗n/p)+∑i=1m/p(p∗ϕ(i∗n))=ϕ(p)∗∑i=1m[gcd(i,p)==1]ϕ(i∗n/p)+∑i=1m/p(ϕ(i/p∗n))+∑i=1m[p|i]((p−1)∗ϕ(i/p∗n))=ϕ(p)∗∑i=1m[gcd(i,p)==1]ϕ(i∗n/p)+∑i=1m/p(ϕ(i/p∗n))+∑i=1m[p|i](ϕ(p)∗ϕ(i/p∗n))=ϕ(p)∑i=1m(i∗n/p)+∑i=1m/pϕ(i∗n) $$\sum_{i=1} ^m\phi(i*n)\\ =\phi(p)*\sum_{i=1} ^m[gcd(i,p)==1]\phi(i*n/p)\\ +\sum_{i=1} ^{m} [p|i]\phi(i*n)\\ =\phi(p)*\sum_{i=1} ^m[gcd(i,p)==1]\phi(i*n/p) +\\\sum_{i=1} ^{m} [p|i](p*\phi(i/p*n))\\=\phi(p)*\sum_{i=1} ^m[gcd(i,p)==1]\phi(i*n/p) +\\\sum_{i=1} ^{m/p} (p*\phi(i*n))\\=\phi(p)*\sum_{i=1} ^m[gcd(i,p)==1]\phi(i*n/p) +\\\sum_{i=1} ^{m/p} (\phi(i/p*n))+\sum_{i=1} ^{m} [p|i]((p-1)*\phi(i/p*n))\\=\phi(p)*\sum_{i=1} ^m[gcd(i,p)==1]\phi(i*n/p) +\\\sum_{i=1} ^{m/p} (\phi(i/p*n))+\sum_{i=1} ^{m} [p|i](\phi(p)*\phi(i/p*n))\\=\phi(p)\sum_{i=1} ^m (i*n/p)+\sum _{i=1} ^{m/p}\phi(i*n)$$