Find the last two digits of 51³⁷.

Step 1. Write the base. $51^{37} = (1 + 50)^{37} = 1 + 37 \times 50 + {}^{37}C_{2} \times 50^2 + \cdots$.

Step 2. Drop what vanishes. $50^2 = 2500$ is a multiple of $100$, so every term from the third onwards is too.

Step 3. Add. $1 + 37 \times 50 = 1851$, and $1851$ leaves remainder $51$ when divided by $100$.