%3Cp%3ELet%20PosZ%26nbsp%3B%3D%26nbsp%3B%7Bz%26nbsp%3B%3F%26nbsp%3BZ%26nbsp%3B%3

Question

%3Cp%3ELet%20PosZ%26nbsp%3B%3D%26nbsp%3B%7Bz%26nbsp%3B%3F%26nbsp%3BZ%26nbsp%3B%3A%26nbsp%3Bz%26nbsp%3B%26gt%3B%26nbsp%3B0%7D.%26nbsp%3BConsider%26nbsp%3Bthe%26nbsp%3Bfunction%26nbsp%3Bf%26nbsp%3B%3A%26nbsp%3BPosZ%26nbsp%3B%3F%26nbsp%3BPosZ%26nbsp%3Bde%3Fned%26nbsp%3Bas%26nbsp%3Bfollows%3A%3C%2Fp%3E%3Cp%3E%E2%80%A2%26nbsp%3Bf(1)%26nbsp%3B%3D%26nbsp%3B1%3C%2Fp%3E%3Cp%3E%E2%80%A2%26nbsp%3BIf%26nbsp%3Bz%26nbsp%3B%3F%26nbsp%3BPosZ%26nbsp%3Band%26nbsp%3Bz%26nbsp%3B%26gt%3B%26nbsp%3B1%26nbsp%3Bthen%26nbsp%3Bf(z)%26nbsp%3Bis%26nbsp%3Bthe%26nbsp%3Blargest%26nbsp%3Binteger%26nbsp%3Bthat%26nbsp%3Bdivides%26nbsp%3Bz%26nbsp%3Bbut%26nbsp%3Bis%26nbsp%3Bdistinct%26nbsp%3Bfrom%26nbsp%3Bz.%26nbsp%3B(For%3C%2Fp%3E%3Cp%3Eexample%26nbsp%3Bf(41)%26nbsp%3B%3D%26nbsp%3B1%26nbsp%3Band%26nbsp%3Bf(36)%26nbsp%3B%3D%26nbsp%3B18.)%3C%2Fp%3E%3Cp%3EProve%26nbsp%3Bthat%26nbsp%3Bf%26nbsp%3Bis%26nbsp%3Bsurjective.%3C%2Fp%3E%3Cdiv%3E%3Cbr%3E%3C%2Fdiv%3E

Explanation / Answer

f(x)=largest integer that divides x.

by seeing this for each positive value of y there should be some y

so f(x) is surjective

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