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Seven different numbers are selected from the integers 1 to 100, and

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cebenez
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Seven different numbers are selected from the integers 1 to 100, and [#permalink] New post   Updated on: 19 Jun 2015, 00:31
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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.

OPEN DISCUSSION OF THIS QUESTION IS HERE: seven-different-numbers-are-selected-from-the-integers-1-to-99943.html

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Official Answer: B

But I think its E, let me tell you why.

(1) Range is 6, I believe when the range is 6, it means the numbers are consecutive. Can range be 6 if not consecutive, I do not think so.

Any set with a range of 6 starting with 3 would be have same sum of remainder but different sum of reminder if starting from 2 i.e for set 2 to 8 is different from set 3 to 9.

So -- Not sufficient.

(2) It says the set is consecutive numbers... Now we do not know the range.. but we know that any set of 7 consecutive numbers > 2 has to have a range of 6. So for the same reasons explained in (1) This is also not sufficient.

1 + 2 ) Together we still have the same info. What if the set starts with 2??? Remainders will be R5, R4, R3, R2, R1, R0, R1... right????

If it starts from 3 to 9... It would be R4, R3, R2, R1, R0, R1, R2

Sum of these remainders are different ... So I think E is the answer....but MGMAT says its B.

MGMT says, (1) is insufficient... but (2) is sufficient because any 7 consecutive nos will have sum of remainders = 21


Can someone explain?
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Official Answer
B

Originally posted by cebenez on 22 Jun 2010, 19:02.
Last edited by Bunuel on 19 Jun 2015, 00:31, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: Seven different numbers are selected from the integers 1 to 100, and [#permalink] New post   22 Jun 2010, 21:37
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What is the sum of the remainders?

1. The range is 6. This actually only means that the highest remainder - lowest remainder is 6
Thus the set could consist of numbers that yield the remainders {0,0,6,6,6,6,6} OR {0,3,4,5,6,6,6} etc. The sum will be different depending on the actual contents of the set. INSUFFICIENT

2. They are consecutive integers. Your example has a few flaws in it:
For 3 to 9 : 3 has a quotient of zero and a remainder of 3
4 has a quotient of zero and a remainder of 4
And so on, yielding remainders {3,4,5,6,0,1,2} TOTAL = 21

For 2 to 8 : The remainders are {2,3,4,5,6,0,1} TOTAL = 21

SUFFICIENT

Pick B.
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Re: Seven different numbers are selected from the integers 1 to 100, and [#permalink] New post   19 Jun 2015, 00:32
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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?

The trick here is to know that remainder is always non-negative integer less than divisor \(0\leq{r}<d\), so in our case \(0\leq{r}<7\).

So the remainder upon division of any integer by 7 can be: 0, 1, 2, 3, 4, 5, or 6 (7 values).

(1) The range of the seven remainders is 6 --> if we pick 6 different multiples of 7 (all remainders 0) and the 7th number 6 (remainder 6) then the range would be 6 and the sum also 6. But if we pick 7 consecutive integers then we'll have all possible remainders: 0, 1, 2, 3, 4, 5, and 6 and their sum will be 21. Not sufficient.

(2) The seven numbers selected are consecutive integers --> ANY 7 consecutive integers will give us all remainders possible: 0, 1, 2, 3, 4, 5, and 6. It does not matter what the starting integer will be: if it's say 11 then the remainder of 7 consecutive integers from 11 divided by 7 will be: 4, 5, 6, 0, 1, 2, and 3 and if starting number is say 14 then the remainder of 7 consecutive integers from 14 divided by 7 will be: 0, 1, 2, 3, 4, 5 and 6. So in any case sum=0+1+2+3+4+5+6=21. Sufficient.

Answer: B.

Similar questions to practice:
n-consecutive-integers-are-selected-from-the-integers-131349.html
seven-integers-x1-x2-x3-x4-x5-x6-and-x7-are-picked-73611.html

OPEN DISCUSSION OF THIS QUESTION IS HERE: seven-different-numbers-are-selected-from-the-integers-1-to-99943.html
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Re: Seven different numbers are selected from the integers 1 to 100, and [#permalink] New post   22 Jun 2010, 23:17
AbhayPrasanna, thanks a lot.

yes you are right. (1) says the range of remainders = 6... I was reading as the range of the set = 6...

(2) yes you are right.. I made some mistake on calculating the remainders... you are right.

Thanks a lot. I am getting a bit nervous, today I have been quite a lot of mistakes, because am not reading the question right. !!! what's happening to me !!!
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Re: Seven different numbers are selected from the integers 1 to 100, and [#permalink] New post   24 Jun 2010, 02:42
Ya..its B..
@ cebenez: You dont need to panic..Sometimes it happens that the logic goes haywire....
Just ensure...its not happening on regular basis...:)
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Re: Seven different numbers are selected from the integers 1 to 100, and [#permalink] New post   24 Jun 2010, 13:27
Answer is B. Here's my explanation.

1. Range is 6.

This statement is insufficient as you can have a range of numbers with six as follows;

{0,0,0,0,0,1,6} or {0,3,4,4,5,5,6}

Range is the difference between the largest and smallest number but that doesn't say anything.

So now we're down to options B,C or E

2. Consecutive numbers.

Since we are choosing 7 consecutive numbers, one of it must be a multiple of 7. So, let's assume its the first number for convenience sake. You can draw a table as follows:

Number Reminder
x - 0
x+1 - 1
x+2 - 2
x+3 - 3
x+4 - 4
x+5 - 5
x+6 - 6

So now you can find out what's asked for.

Hence answer is B.



cebenez wrote:
Question 98 Data Sufficiency Official Guide

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.


Official Answer: B

But I think its E, let me tell you why.

(1) Range is 6, I believe when the range is 6, it means the numbers are consecutive. Can range be 6 if not consecutive, I do not think so.

Any set with a range of 6 starting with 3 would be have same sum of remainder but different sum of reminder if starting from 2 i.e for set 2 to 8 is different from set 3 to 9.

So -- Not sufficient.

(2) It says the set is consecutive numbers... Now we do not know the range.. but we know that any set of 7 consecutive numbers > 2 has to have a range of 6. So for the same reasons explained in (1) This is also not sufficient.

1 + 2 ) Together we still have the same info. What if the set starts with 2??? Remainders will be R5, R4, R3, R2, R1, R0, R1... right????

If it starts from 3 to 9... It would be R4, R3, R2, R1, R0, R1, R2

Sum of these remainders are different ... So I think E is the answer....but MGMAT says its B.

MGMT says, (1) is insufficient... but (2) is sufficient because any 7 consecutive nos will have sum of remainders = 21


Can someone explain?
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Re: Seven different numbers are selected from the integers 1 to 100, and [#permalink] New post   24 Jun 2010, 18:11
Just a quick note that I thought of when looking at the problem.

(1) is actually something we already know given the question. The range of the remainder of any number divided by 7 is going to be 6. The remainder of any number divided by seven will be between 0-6. A remainder of 7 or more would need to be adjusted so that the appropriate number is added to the quotient leaving only a remainder that within the range of 0-6.

If you caught this you can quickly dismiss (1) as redundant information and move on to (2). This can save time for those of you who tried to go about (1) by testing numbers or doing other mathematical exercises.

Hope this helps.

Thanks,

Jared
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Re: Seven different numbers are selected from the integers 1 to 100, and [#permalink]
24 Jun 2010, 18:11
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