lheiannie07 wrote:Seven different numbers are selected from the integers 1 to 100, and each number is divided by 7. What is the sum of the remainders?
(1) The range of the seven remainders is 6.
(2) The seven numbers selected are consecutive integers.
Can some experts explain why option E isn't the best option?
OA B
We have to randomly choose 7 different integers from 1 to 100 and get the sum of the remainders when they are divided by 7.
Let's take each statement one by one.
(1) The range of the seven remainders is 6.
Important here to note that the range of the seven
remainders is 6.
Case 1: Say the 7 different integers are 7, 13, 14, 21, 28, 35, and 42
The respective reminders are 0, 6, 0, 0, 0, 0, and 0. This case ensures that the range of the remainders = 6 - 0 = 0.
The sum of the reminders = 0+6+0+0+0+0+0 = 6.
Case 2: Say the 7 different integers are 7, 13, 15, 22, 29, 36, and 43
The respective reminders are 0, 6, 1, 1, 1, 1, and 1. This case ensures that the range of the remainders = 6 - 0 = 0.
The sum of the reminders = 0+6+1+1+1+1+1 = 11.
No unique answer. Insufficient.
(2) The seven numbers selected are consecutive integers.
Since there are 7 consecutive integers, one of them would be divisible by 7, leaving remainder = 0, and the other 6 will leave remainders 1, 2, 3, 4, 5, and 6. The order will change but the set of remainders would ALWAYS be {0, 1, 2, 3, 4, 5, 6}.
Thus, the sum of the reminders = 0+1+2+3+4+5+6 = 21
Let's take a couple of cases to get this better.
Case 1: Say the consecutive 7 integers are 10, 11, 12, 13, 14, 15 and 16
The respective remainders are 3, 4, 5, 6, 0, 1, and 2. We see their sum = 0+1+2+3+4+5+6 = 21.
Case 2: Say the consecutive 7 integers are 12, 13, 14, 15, 16, 17 and 18
The respective remainders are 5, 6, 0, 1, 2, 3, and 4. We see their sum = 0+1+2+3+4+5+6 = 21.
Unique answer. Sufficient
The correct answer:
B
Hope this helps!
-Jay
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