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# Help! A remainder is chasing me

Best Answer normdeplume , 19 September 2007 - 07:23 PM

My answer is below, is this the answer you were looking for or is there a lower one?

Spoiler for solution

Just because your voice reaches halfway around the world doesn't mean you are wiser than when it reached only to the end of the bar. Go to the full post

67 replies to this topic

### #61 Faizal

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Posted 08 January 2012 - 09:07 AM

the smallest we can find is 2519
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### #62 Nep Ton

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Posted 12 January 2012 - 01:20 AM

surely the correct answer on my account would be ... NO
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### #63 dheeraj

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Posted 25 May 2012 - 07:51 AM

Spoiler for doesnt a smaller number work?

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### #64 OMGTHATSIT

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Posted 06 June 2012 - 12:24 AM

if you check out each number out each number's posiblities and put it into a formula, you will obtain:

for the number 2: (x is any natural number, and it's diferent for every equation)

1+ 2x

Number 3

29+30x

Number 4

19+ 20x

Number 5

9+ 10x

Number 6

29 + 30x

Number 7

69 + 70x

Number 8

39 + 40x

Number 9

89+90x

Number 10

9 + 10X

So we need a number that is equivalent to all those equations. we have:

1+2X

29+30X

19+20X

9+10X

69+70X

39+40x

89+90X

So, we go and analyse the equations who give bigger products for the smallest number of X. those are: 89+90x and 69 + 70x. Replacing one of the x for another letter (y) we get:

89+90y = 69 + 70x <=> 20 = 70x - 90y

With that, we can now replace the "x" and the "y". for that formula to work, after some calculations, I notice that the x must go for 8 + 9z and the y must go 6+7z

So, starting for the 1st, if z = 0, x=8 and y=6. we replace them in the equation

89 + 90 x 6 = 629.

However, this number doesnt exist on the equation 39+40x, so its invalid.

You keep going,

z=1; x= 17 and y = 13, the final =1259, also doesnt exist on the equation 39+40x, so its invalid.

z=2, the final is 1889, also invalid.

Then, finally, z=43 y = 6+(7x3) =27........ 89 + 90x27 = 2519; which is valid for all the equations.
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### #65 Jonne

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Posted 06 June 2012 - 02:34 PM

I actually believe I've found a smaller number, if negative numbers are allowed.
-1 !

-1 = -1 * 10 + 9
-1 = -1 * 9 + 8
-1 = -1 * 8 + 7
-1 = -1 * 7 + 6
-1 = -1 * 6 + 5
-1 = -1 * 5 + 4
-1 = -1 * 4 + 3
-1 = -1 * 3 + 2
-1 = -1 * 2 + 1
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### #66 brifri238

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Posted 06 November 2012 - 04:05 PM

For those of those familiar with SQL, these types of problems are easily solved by perfoming a query on a Numbers (or Tally) table.

For example I have run the following selection on a numbers table (Numbers) containing 100,000,000 Rows (1,2,3....100,000,000) for the first 17 rows of the sequence (2 remainer 1 up to 18 remaining 17)

SELECT
Min(Number)
FroM Numbers
WHERE
Number%2 = 1
AND Number%3 = 2
AND Number%4 = 3
AND Number%5 = 4
AND Number%6 = 5
AND Number%7 = 6
AND Number%8 = 7
AND Number%9 = 8
AND Number%10 =9
AND Number%11 = 10
AND Number%12 = 11
AND Number%13 = 12
AND Number%14 = 13
AND Number%15 = 14
AND Number%16 = 15
AND Number%17 = 16
AND Number%18 = 17

(To get the solutions to next rows in the sequence: 19 remainder 18, 20 remainder 19...) requires a bigger Number table

Edited by brifri238, 06 November 2012 - 04:06 PM.

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### #67 brifri238

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Posted 06 November 2012 - 05:31 PM

FYI: If you extend the sequence from 2 remainder 1 .... to bigger numbers

possible solutions are (I am not absoutely sure they are the smallest number that works for each but these work!)

20 remainder 19 ......................232,792,559
22 remainder 21.......................232,792,559
23 remainder 22....................5,354,228,879
24 remainder 23....................5,354,228,879

30 remainder 29.............2,329,089,562,799
31 remainder 30...........72,201,776,446,799
36 remainder 35.........144,403,552,893,599
40 remainder 39......5,342,931,457,063,199

Edited by brifri238, 06 November 2012 - 05:35 PM.

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### #68 timfoden

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Posted 18 January 2014 - 09:23 AM

Spoiler for doesnt a smaller number work?

Spoiler for Sorry, I'm afraid not...

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