Math Snap

STEP 1

1. We are looking for the remainder when $17^{201}$ is divided by 18.
2. We can use properties of modular arithmetic to simplify the problem.

STEP 2

1. Simplify the base using modular arithmetic.
2. Use properties of exponents in modular arithmetic.
3. Calculate the remainder.

STEP 3

Simplify the base $17$ modulo $18$.
$$17 \equiv -1 \pmod{18}$$

STEP 4

Use the property of exponents in modular arithmetic:
$$17^{201} \equiv (-1)^{201} \pmod{18}$$

STEP 5

Calculate $(-1)^{201}$.
Since 201 is an odd number:
$$(-1)^{201} = -1$$ Thus:
$$(-1)^{201} \equiv -1 \pmod{18}$$

Convert $-1$ to a positive remainder:
$$-1 \equiv 17 \pmod{18}$$ The remainder when $17^{201}$ is divided by 18 is:
$$\boxed{17}$$