Here is a similar example: Suppose you want to find x where (0 ≤ x ≤ 100) and 17¹³ ≡ x mod 101. As 17¹³ is too large for most calculators to show exactly we start with 17⁶=24137569 and, first dividing this by 101, we find that 17⁶=(238985)(101)+84 so we now know that 17⁶ ≡ 84 mod 101.

The next step is to use this to tackle 17¹³.

1713 =(176)2 ×17 ≡ 842 ×17 ≡ 119952 mod 101 119952 =1187×101 + 65 ≡ 65 mod 101.

Hence x=65.