How did Euler help establish Fermat's little theorem?

Although Pierre de Fermat first stated his "little theorem" in 1640, he left no known proof for the assertion that for any prime $p$ and any integer $a$ not divisible by $p$, $p$ divides $a^p-a$.1

The Swiss mathematician Leonhard Euler is credited with providing the first known published proof of this theorem in 1736. Euler’s work on the theorem was part of a broader effort to bring number theory into the mathematical mainstream during the 18th century. Encouraged by his correspondence with Christian Goldbach, Euler dedicated decades to researching and rigorously proving many of Fermat’s previously unverified observations. While Gottfried Wilhelm Leibniz had also produced a proof in an unpublished manuscript around 1683, Euler’s 1736 publication was the first to formally establish the result in the scientific record.12