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# Seven different numbers are selected from the integers 1 to

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Manager
Joined: 17 Mar 2010
Posts: 171

Kudos [?]: 211 [0], given: 9

Seven different numbers are selected from the integers 1 to [#permalink]

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01 Aug 2010, 10:39
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Difficulty:

(N/A)

Question Stats:

67% (00:33) correct 33% (00:29) wrong based on 12 sessions

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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: http://gmatclub.com/forum/seven-differe ... 99943.html
[Reveal] Spoiler: OA

Kudos [?]: 211 [0], given: 9

Senior Manager
Status: Time to step up the tempo
Joined: 24 Jun 2010
Posts: 404

Kudos [?]: 259 [1], given: 50

Location: Milky way
Schools: ISB, Tepper - CMU, Chicago Booth, LSB
Re: Seven different numbers are selected from the integers 1 to [#permalink]

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01 Aug 2010, 16:43
1
KUDOS
Here is my attempt.

Statement A: The range only tells us the spread between the highest and the lowest. It is possible to have different values for remainders and still have the same range.

i.e., remainders could be 0,6,6,6,6,6 or 0,6, a, b, c, d, e with a, b, c, d and e being any integer between 0 and 6.

Hence statement A is insufficient

Statement B: Seven consecutive numbers would definitely have one number that is a multiple of 7 and other numbers would contribute a remainder that is less than 7.

Numbers could be 7x, 7x+1, 7x+2, ...... 7x+6

Hence statement B is sufficient.
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Kudos [?]: 259 [1], given: 50

Math Expert
Joined: 02 Sep 2009
Posts: 42259

Kudos [?]: 132713 [0], given: 12335

Re: Seven different numbers are selected from the integers 1 to [#permalink]

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14 Aug 2017, 02:07
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.

Similar questions to practice:
http://gmatclub.com/forum/n-consecutive ... 31349.html
http://gmatclub.com/forum/seven-integer ... 73611.html

OPEN DISCUSSION OF THIS QUESTION IS HERE: http://gmatclub.com/forum/seven-differe ... 99943.html
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Kudos [?]: 132713 [0], given: 12335

Re: Seven different numbers are selected from the integers 1 to   [#permalink] 14 Aug 2017, 02:07
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