From 1 to 100

Number Theory Level 3

True or False?

There exist distinct positive integers $a$ and $b$ such that $a^{100}+b^{100}$ is divisible by all of the following numbers: $a+b, a^2+b^2, a^3+b^3, a^4+b^4, \ldots, a^{99}+b^{99}, a^{100}+b^{100}.$