Bunuel wrote:
Competition Mode Question
A woman who has two sons enrolled in an elementary school brings a batch of n cookies to the school. Her youngest son is in a class with 10 other students and her oldest son is in a class with 7 other students. If she divides the n cookies evenly among the students in her youngest son’s class there will be 5 left over. If she divides the cookies evenly among the students in her oldest son’s class there will be 7 left over. If she has more than 200 cookies, which of the following is the sum of the digits of the smallest possible value of n?
A. 9
B. 11
C. 13
D. 17
E. 23
Are You Up For the Challenge: 700 Level QuestionsWe see that there are 11 students in her younger son’s class and 8 students in her older son’s class. We are given than n is greater than 200. Furthermore, we see that when n is divided by 11, the remainder is 5 and when n is divided by 8, the remainder is 7.
Even though we know that n is greater than 200, let’s start by listing the numbers (a lot smaller than 200) that leave a remainder of 5 when divided by 11:
5, 16, 27, 38, 49, 60, 71, …
We see that 71 is the smallest number that leaves a remainder of 5 when divided by 11 and also a remainder of 7 when divided by 8 (notice that 71/11 = 6 R 5 and 71/8 = 8 R 7). Now, to get the smallest value of n, we can keep adding the LCM of 11 and 8 (i.e., 88) to 71 until it becomes more than 200:
71 + 88 = 159
159 + 88 = 247
Therefore, the smallest value of n is 247, and the sum of the digits of 247 is 2 + 4 + 7 = 13.
Answer: C
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