bonanova 81 Report post Posted September 19, 2007 I just found a number with an interesting property: When I divide it by 2, the remainder is 1. When I divide it by 3, the remainder is 2. When I divide it by 4, the remainder is 3. When I divide it by 5, the remainder is 4. When I divide it by 6, the remainder is 5. When I divide it by 7, the remainder is 6. When I divide it by 8, the remainder is 7. When I divide it by 9, the remainder is 8. When I divide it by 10, the remainder is 9. It's not a small number, but it's not really big, either. When I looked for a smaller number with this property I couldn't find one. Can you find it? Share this post Link to post Share on other sites

0 Guest Report post Posted September 19, 2007 My answer is below, is this the answer you were looking for or is there a lower one? 2519 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. 1 Share this post Link to post Share on other sites

0 bonanova 81 Report post Posted September 19, 2007 That's it. Here's my method: [care to share yours?] The number has to end in 9. Looked brute force for small numbers. 59 and 119 were promising, but no cigar. Then looked for agreement among 39 + multiples of 40, 69 + multiples of 70 and 89 + multiples of 90 Smallest one was 2519. Still think of this as kind of brute force. Maybe there is no elegant solution. Share this post Link to post Share on other sites

0 Guest Report post Posted September 20, 2007 I am afraid, it was pretty much the 'brute force' method. As you say 59 and 119 looked promising, both working upto and including 6, but 7 was the dificulty, so I tried each increment of 60 (179, 239, 299 ...), until I got to 419, which worked for 7 as well, (hooray) but not 9 or 10 (boo). Then using the same logic as before (59 being the first number that worked up the number 6, Increments of 60 always worked for the first 6 numbers. So 419 being the first that works for numbers up to 7 I incremented by 420.) So 419, 839, 1259, 1679, 2099, 2519. Not a very elogant solution, but it passed a little time. I assume, some mathemarical genius is going to post an easier way 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 Share this post Link to post Share on other sites

0 Guest Report post Posted October 3, 2007 If we're looking for n where n divided by x (x being each number 2-10) leaves a remainder of x-1, n+1 divided by each number 2-10 leaves no remainder. To find n+1, we must simply multiply all prime factors of 2-10, excluding any duplicates. This is equal to 2*2*2*3*3*5*7 = 2520, then just subtract one to get the answer of 2519. 1 Share this post Link to post Share on other sites

0 Guest Report post Posted October 3, 2007 I just found a number with an interesting property: When I divide it by 2, the remainder is 1. When I divide it by 3, the remainder is 2. When I divide it by 4, the remainder is 3. When I divide it by 5, the remainder is 4. When I divide it by 6, the remainder is 5. When I divide it by 7, the remainder is 6. When I divide it by 8, the remainder is 7. When I divide it by 9, the remainder is 8. When I divide it by 10, the remainder is 9. It's not a small number, but it's not really big, either. When I looked for a smaller number with this property I couldn't find one. Can you find it? Let X be the number. From the above properties of X, one can see that, "X+1" is a multiple of the numbers from 2 to 10. As Bonanova is asking for the smallest number with such properties, "X+1" must be the LCM (least common multiple) of the numbers from 2 to 10 -- which is 2520. And hence, X = 2520-1 = 2519. 1 Share this post Link to post Share on other sites

0 Guest Report post Posted October 3, 2007 Another variation of the above problem is as follows: I just found a number with an interesting property: When I divide it by 2, the remainder is 1. When I divide it by 3, the remainder is 1. When I divide it by 4, the remainder is 1. When I divide it by 5, the remainder is 1. When I divide it by 6, the remainder is 1. When I divide it by 7, the remainder is 1. When I divide it by 8, the remainder is 1. When I divide it by 9, the remainder is 1. When I divide it by 10, the remainder is 1. But when I divide it by 11, the remainder is 0 (just to make the procedure a little bit different). Can you find it? Share this post Link to post Share on other sites

0 Guest Report post Posted October 6, 2007 Not sure if people are checking the variation of the above question. I think it is better to post it as a new question. Share this post Link to post Share on other sites

0 Guest Report post Posted January 15, 2008 All we need to do is to take the LCM of all these numbers and subtract 1 from that.. LCM of 1 to 10 is 5x8x9x7=2520 -1 is 2519 Share this post Link to post Share on other sites

0 Guest Report post Posted January 18, 2008 Another variation of the above problem is as follows: I just found a number with an interesting property: When I divide it by 2, the remainder is 1. When I divide it by 3, the remainder is 1. When I divide it by 4, the remainder is 1. When I divide it by 5, the remainder is 1. When I divide it by 6, the remainder is 1. When I divide it by 7, the remainder is 1. When I divide it by 8, the remainder is 1. When I divide it by 9, the remainder is 1. When I divide it by 10, the remainder is 1. But when I divide it by 11, the remainder is 0 (just to make the procedure a little bit different). Can you find it? A number that satisfies all of these properties is: 25201 Share this post Link to post Share on other sites

0 Guest Report post Posted January 22, 2008 I'de write a computer program in C# that has 10 ifs and get to the number very quickly, but without the mathematic understanding, and for that I thank you guys Share this post Link to post Share on other sites

0 Guest Report post Posted January 30, 2008 wrote a simple script - the smallest number is 2519. here is my way. $solution = "F"; $x=0; while($solution eq "F") { $x++; $flag=0; for($i=1;$i<10;$i++) { $j=$i+1; $y=$x%$j; if($y != $i) {$flag=1;} } if($flag==0) { $solution="T"; print $x,"\n"; } } ## it is perl. Share this post Link to post Share on other sites

0 Guest Report post Posted January 31, 2008 wrote a simple script - the smallest number is 2519. here is my way. $solution = "F"; $x=0; while($solution eq "F") { $x++; $flag=0; for($i=1;$i<10;$i++) { $j=$i+1; $y=$x%$j; if($y != $i) {$flag=1;} } if($flag==0) { $solution="T"; print $x,"\n"; } } ## it is perl. That seems to take more knowledge of the working mathematics than the other mathematics solutions provided. Guess it just goes to show that math and logic are very much the same animal. Share this post Link to post Share on other sites

0 Guest Report post Posted February 2, 2008 Actually any number (2520 * K) - 1 where k is any positive integer has these properties Share this post Link to post Share on other sites

0 Guest Report post Posted February 13, 2008 This problem is simple. All you have to do is find the least common multiple of 2,3,4,5,6,7,8,9, and,10. Then you subtract 1 from that number so that it is not divisible by anyone of those numbers. the number is 2519 Share this post Link to post Share on other sites

0 Guest Report post Posted February 14, 2008 Can also just use XL VB...took 30 secs...just use "mod" to check the remainder, and see if they add up to 45... Sub xxxx() For i = 1 To 100000 Cells(1, 3) = i If Cells(10, 1) = 45 Then Exit For End If Next i End Sub Share this post Link to post Share on other sites

0 bonanova 81 Report post Posted February 15, 2008 I'm curious ... any of you programmers ever use APL? It's interpretive, very useful, and very ancient. Share this post Link to post Share on other sites

0 Guest Report post Posted February 15, 2008 I'm curious ... any of you programmers ever use APL? It's interpretive, very useful, and very ancient. Please, you're really showing your age. :-) After reading the Wikipedia entry and looking at those samples, I'd have to agree that it looks really cool, and totally impractical for any of the work I do. I like the poem at the bottom of the page: 'Tis the dream of each programmer Before his life is done, To write three lines of APL And make the damn thing run. Perhaps I can make some time on the weekends to learn an arcane, remarkably dense language that I'll never use in real life for the purpose of solving math problems with less code. That I say this in all seriousness is probably a rather sad commentary on my social life. Share this post Link to post Share on other sites

0 Guest Report post Posted February 15, 2008 I just found a number with an interesting property: When I divide it by 2, the remainder is 1. When I divide it by 3, the remainder is 2. When I divide it by 4, the remainder is 3. When I divide it by 5, the remainder is 4. When I divide it by 6, the remainder is 5. When I divide it by 7, the remainder is 6. When I divide it by 8, the remainder is 7. When I divide it by 9, the remainder is 8. When I divide it by 10, the remainder is 9. It's not a small number, but it's not really big, either. When I looked for a smaller number with this property I couldn't find one. Can you find it? No need of a brute force method... Just take an LCM (least common multiple) of all the numbers in question and subtract 1 from it! so the LCM of 2,3,4,5,6,7,8,9,10 = 8x9x7x5 = 2520 Hence, the solutions is 2519! Share this post Link to post Share on other sites

0 TwoaDay 0 Report post Posted February 15, 2008 2519 Share this post Link to post Share on other sites

0 Guest Report post Posted February 15, 2008 (edited) In the spirit of ATL, I present to you my method, in C#, for calculating the answer by brute force, without taking into account LCM's. I'm curious how much smaller this could be done. int GetNumber() { int i=1, n=1; bool r=false; for (;!r;){if(i%n==(n-1)){n=n%11+1;r=n==11;}else{i++;n=1;}}return i;} Edited February 15, 2008 by Duh Puck Share this post Link to post Share on other sites

0 Guest Report post Posted March 9, 2008 I just found a number with an interesting property: When I divide it by 2, the remainder is 1. When I divide it by 3, the remainder is 2. When I divide it by 4, the remainder is 3. When I divide it by 5, the remainder is 4. When I divide it by 6, the remainder is 5. When I divide it by 7, the remainder is 6. When I divide it by 8, the remainder is 7. When I divide it by 9, the remainder is 8. When I divide it by 10, the remainder is 9. It's not a small number, but it's not really big, either. When I looked for a smaller number with this property I couldn't find one. Can you find it? I think you should include dividing 11 to get a remainder of 10 and dividing 12 to get a remainder of 11 Share this post Link to post Share on other sites

0 Guest Report post Posted March 9, 2008 (edited) It's the least common multiple of 2, 3, 4, 5, 6, 7, 8, 9, and 10, then minus one, which is 2x2x2x3x3x5x7 minus one, which is 2519. Edited March 9, 2008 by babbal Share this post Link to post Share on other sites

0 Guest Report post Posted March 17, 2008 wrote a simple script - the smallest number is 2519. here is my way. $solution = "F"; $x=0; while($solution eq "F") { $x++; $flag=0; for($i=1;$i<10;$i++) { $j=$i+1; $y=$x%$j; if($y != $i) {$flag=1;} } if($flag==0) { $solution="T"; print $x,"\n"; } } ## it is perl. Do people actually use perl? I never have.. In the spirit of ATL, I present to you my method, in C#, for calculating the answer by brute force, without taking into account LCM's. I'm curious how much smaller this could be done. int GetNumber() { int i=1, n=1; bool r=false; for (;!r;){if(i%n==(n-1)){n=n%11+1;r=n==11;}else{i++;n=1;}}return i;} Well, that is shorter code than mine would be, but mine would be done sooner. I don't feel like writing it, but I could do it! I am actually kinda disappointed in myself for not realizing it was a LCM problem... when I was thinking about it in my head, I was considering each sentence seperately and finding a rule for each one. First there was no even numbers allowed (2:1), then the last digit had to be a 9 (10:9), then the second to last digit had to be one less than 3, 6, or 9 (3:2), etc. I came up with 5039 this way... (5039+1)/2 - 1 = 2519, go figure Share this post Link to post Share on other sites

0 Guest Report post Posted March 18, 2008 I really enjoyed this one, as it pressed my math brain a little to figure out the best way to solve it. Alas, I believe I found the answer (assuming it must be positive, of course) ... 2519 Share this post Link to post Share on other sites

0 Guest Report post Posted March 18, 2008 Another variation of the above problem is as follows: I just found a number with an interesting property: When I divide it by 2, the remainder is 1. When I divide it by 3, the remainder is 1. When I divide it by 4, the remainder is 1. When I divide it by 5, the remainder is 1. When I divide it by 6, the remainder is 1. When I divide it by 7, the remainder is 1. When I divide it by 8, the remainder is 1. When I divide it by 9, the remainder is 1. When I divide it by 10, the remainder is 1. But when I divide it by 11, the remainder is 0 (just to make the procedure a little bit different). Can you find it? This time X-1 should be the LCM and hence the number is 2521 Share this post Link to post Share on other sites

I just found a number with an interesting property:

When I divide it by 2, the remainder is 1.

When I divide it by 3, the remainder is 2.

When I divide it by 4, the remainder is 3.

When I divide it by 5, the remainder is 4.

When I divide it by 6, the remainder is 5.

When I divide it by 7, the remainder is 6.

When I divide it by 8, the remainder is 7.

When I divide it by 9, the remainder is 8.

When I divide it by 10, the remainder is 9.

It's not a small number, but it's not really big, either.

When I looked for a smaller number with this property I couldn't find one.

Can you find it?

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