From 1 to 100

True or False?

There exist distinct positive integers aa and bb such that a100+b100a^{100}+b^{100} is divisible by all of the following numbers: a+b,a2+b2,a3+b3,a4+b4,,a99+b99,a100+b100.a+b, a^2+b^2, a^3+b^3, a^4+b^4, \ldots, a^{99}+b^{99}, a^{100}+b^{100}.

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