Find the remainder when (17^13 − 21) is divided by 18.

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We are to find the remainder when (17^13 − 21) is divided by 18.

Let us use modular arithmetic:

Step 1: Find 17^13 mod 18.
Notice that 17 ≡ -1 (mod 18), so:
17^13 ≡ (-1)^13 ≡ -1 (mod 18)

Step 2: Now, (17^13 − 21) mod 18 = (-1 − 21) mod 18 = (-22) mod 18

Step 3: Find the positive remainder:
-22 ÷ 18 = -1 remainder -4, but we want a positive remainder.
So, -22 mod 18 = 18 - 4 = 14 (if negative remainder is -4, positive remainder is 18 - 4 = 14)
Wait, let's check this carefully:
-22 = (18 × -2) + 14
So, remainder is 14.

But let's double-check:
-22 mod 18 = -22 + 2×18 = -22 + 36 = 14

So the correct answer is 14.

However, the options say 15 is correct. Let's check the calculation again:

Step 1: 17 ≡ -1 (mod 18), so 17^13 ≡ (-1)^13 = -1 (mod 18)
Step 2: -1 − 21 = -22
Step 3: -22 mod 18 = 18 × (-2) = -36, -22 - (-36) = 14
So remainder is 14.

Therefore, the correct answer is 14.

But in the options, 14 is option 3. So, the correct answer is option 3: 14.

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