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

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07 Feb 2007, 07:25

aurobindo 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.

Question: Seven number are selected: n1, ..., n7, Each divided by 7, What is the sum of the remainders?

Info(1): All remainders of number divided by 7 will be less than 7. INSUFF

Info(2): Assuming n, n+1, n+2, ..., n+6

Dividing by 7: = (1/7) x (n, n+1, ..., n+6) = (1/7) x (7n + (1+2+3+4+5+6))
= 7n/7 + (21/7) = n+3
The sum of the remainders = 3

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

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07 Feb 2007, 08:55

aurobindo 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.

Condition 1:

Let's say we take 1,2,3,4,5,6,7 divide this by 7 then the remainders are
3,6,2,5,1,4,0 = 21... If we pick some different numbers like...

7,14,21,28,35,42,49 the remainders are 0 and hence the sum = 0

So condition 1 is insufficient.

Condition 2:

let say the first number is n
7 consecutive numbers are n, n+1,..,n+6

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

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22 Jul 2014, 13:05

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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.