2018 AMC 8 Problem 7

Attempt Problem 7 of the 2018 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 8 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

7.

The 55-digit number 2\underline{2} 0\underline{0} 1\underline{1} 8\underline{8} U\underline{U} is divisible by 9.9. What is the remainder when this number is divided by 8?8?

1 1

3 3

5 5

6 6

7 7

Answer: B
Concepts:divisibilitymodular arithmetic
Difficulty rating: 960
Video solution:
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Written solution:

Notice that a number is divisible by 99 if and only if the sum of its digits is also divisible by 9.9.

The sum of the digits of the 5-digit number in the problem is: 2+0+1+8+U=11+U.2+0+1+8+U= 11+U. As 2\underline{2} 0\underline{0} 1\underline{1} 8\underline{8} U\underline{U} is divisible by 9,9, 11+U11+U must also be divisible by 9.9. Also, as UU is a digit, we know that 0U9.0\le U\le 9. This means that UU can only be 7.7.

Now we know that the 5-digit number in question is 20187, and we want to find the remainder when we divide 20187 by 8. To solve this, simply use long division to see that 20187=25238+3.20187=2523\cdot 8 + 3. Therefore, the remainder is 3.3.

Thus, the correct answer is B.

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