2520 A wonderfull little number

mikau · 2006-02-02 13:55:44

2520 is my favorite number. I found her one day when I decided to find the least common multiple of the numbers 1 through 10 and we've been friends ever since!

2 * 2 * 2 * 3 * 3 * 5 * 7 = 2520

Not only is this number divisible by 1 through 10, it is divisible by any product of the above factors. Which produce the following:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72, 84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630, 840, 1260, 2520,

48 numbers total!

You can also multiply 2 * 2 * 2 * 2 * 3 * 3 * 5 * 7 * 11 * 13 * 17 * 19 to get 232,792,560, a number divisible by 1 through 20, and every product of these numbers. But this number is large and hard to remember, not nearly as sexy as 2520. Relatively small, and easy to remember! :-D

Last edited by mikau (2006-02-02 13:56:13)

mikau · 2006-02-02 14:10:12

To quickly find all the numbers 2520 can be evenly divided by, I wrote a simple program to check divisibilty for me. Its simple but pretty cool.

#include <iostream>

int main()
{

  long int counter = 0;
  float F;
  long int I;
  long int N = 2520;

  for (float i = 1; i <= N; i++)
  {

    F = N/i;
    I = F;


    if (I == F) { std::cout << " " << i << ","; counter++;}

  }

std::cout << "\n\n  " << counter << " found\n";

  return 0;

}


}

It simply checks the divisibility of N by dividing by every integer from 1 to N. Each time the quotient is assigned to a float variable "F" which strores a decimal number. Then an integer variable I is assigned the value of F. In C++, if a decimal number is assigned to an integer, the value will be truncated (basicly anything behind the decimal point is "chopped off"). The program then checks to see if I and F are equal. If they are, that means the integer was not truncated, which only happens if the float value was an integer. Which would only happen if N was divided by a divisble number. So if F and I are equal, the number checked is printed. :-)

I tried assigning N a value of 232,792,560 but for some reason it didn't work. Just spat out every number from between 1 and N and crashed hafway. Maybe too large a number...

mikau · 2006-02-02 14:22:31

2*2*2*2*3*3*5*7*11*13 = 720,720 (Least common multiple of 1 throuh 16.) divisible by:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48, 52, 55, 56, 60, 63, 65, 66, 70, 72, 77, 78, 80, 84, 88, 90, 91, 99, 104, 105, 110, 112, 117, 120, 126, 130, 132, 140, 143, 144, 154, 156, 165, 168, 176, 180, 182, 195, 198, 208, 210, 220, 231, 234, 240, 252, 260, 264, 273, 280, 286, 308, 312, 315, 330, 336, 360, 364, 385, 390, 396, 420, 429, 440, 455, 462, 468, 495, 504, 520, 528, 546, 560, 572, 585, 616, 624, 630, 660, 693, 715, 720, 728, 770, 780, 792, 819, 840, 858, 880, 910, 924, 936, 990, 1001, 1008, 1040, 1092, 1144, 1155, 1170, 1232, 1260, 1287, 1320, 1365, 1386, 1430, 1456, 1540, 1560, 1584, 1638, 1680, 1716, 1820, 1848, 1872, 1980, 2002, 2145, 2184, 2288, 2310, 2340, 2520, 2574, 2640, 2730, 2772, 2860, 3003, 3080, 3120, 3276, 3432, 3465, 3640, 3696, 3960, 4004, 4095, 4290, 4368, 4620, 4680, 5005, 5040, 5148, 5460, 5544, 5720, 6006, 6160, 6435, 6552, 6864, 6930, 7280, 7920, 8008, 8190, 8580, 9009, 9240, 9360, 10010, 10296, 10920, 11088, 11440, 12012, 12870, 13104, 13860, 15015, 16016, 16380, 17160, 18018, 18480, 20020, 20592, 21840, 24024, 25740, 27720, 30030, 32760, 34320, 36036, 40040, 45045, 48048, 51480, 55440, 60060, 65520, 72072, 80080, 90090, 102960, 120120, 144144, 180180, 240240, 360360, 720720,

240 found. :-)

Last edited by mikau (2006-02-02 14:25:42)

MathsIsFun · 2006-02-03 11:01:12

Fascinating stuff!

720,720 seems even more magical because of the repeated numbers.

I wonder what 2520 would look like in other bases?

mikau · 2006-02-03 13:15:11

Yeah 2520 is the LCM of the numbers 1 through 10. For 1 through 11 we need an addition factor of 11 which brings it up to 27,720. This is also the lcm of 1 to 12 since we already have the factors of 12 present. For 1 to 13 we need an additional factor of 13 which brings it up to 360,360. The factors of 14 and 15 are already present so this is also the lcm of 1 to 15. For a factor of 16 we need an additional factor of 2, (we only have three) that brings it up to 720,720.

God · 2006-02-05 14:34:13

100110100000 in binary, 9D8 in hex, 10230 in 7..... that's all I care about

IMHO 65536 is way cooler than 2520.
65536 = 2^2^2^2. That's 2 to it's own power 2^2 times.

mikau · 2006-02-28 07:04:28

Well I've addapted 720,720 as my new favorite number. Originally I didn't investigate any further then 4 digits as it became hard to remember, but 720,720 is easy to remember!

krassi_holmz · 2006-02-28 07:11:16

You can compute the number
p_1p_2..p_n, which will be divisible to all the numbers less than p_n+1.

krassi_holmz · 2006-02-28 07:18:50

The number 30030 is divisible to all the numbers less than 17 and it's easy to remember.

krassi_holmz · 2006-02-28 07:24:08

Othen easy-to-remember numbers are:
510510 which is divisible to all numbes less than 19
9699690 -//- to all<23
Up to
125468249177422704873057441198265167787559733358158169962944986266696409522664\
838749597773492757291842527159639796689556759017016285579548286320186227272766\
165197163269594602008771743253300924706081546906873165900298789082674560906079\
856481025298916728267733098788961332559413072403989323365304158393895684891063\
875936658095458174029029411209474267592081293674698128271264634147613494631818\
195639290031311461367704339650484668757761790715896528147421600075132164763634\
038186948603982465875980085049712183867633467531972781913315834804381159682635\
298479358050289946346623166324630256580401063842526350008401664018893489414182\
30630291758422303205747023179473557711721130190
it doesn't seem to be other "easy-to-remember" number.

mikau · 2006-03-01 09:46:29

sweet stuff!

krassi_holmz · 2006-03-03 09:52:00

ya, swweett

Monox D. I-Fly · 2019-11-14 14:31:31

mikau wrote:

2520 is my favorite number. I found her one day when I decided to find the least common multiple of the numbers 1 through 10 and we've been friends ever since!
2 * 2 * 2 * 3 * 3 * 5 * 7 = 2520
Not only is this number divisible by 1 through 10, it is divisible by any product of the above factors. Which produce the following:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72, 84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630, 840, 1260, 2520,
48 numbers total!
You can also multiply 2 * 2 * 2 * 2 * 3 * 3 * 5 * 7 * 11 * 13 * 17 * 19 to get 232,792,560, a number divisible by 1 through 20, and every product of these numbers. But this number is large and hard to remember, not nearly as sexy as 2520. Relatively small, and easy to remember! :-D

Glad the ancient people didn't realize this. Otherwise, 1 full circle will be treated as 2,520° instead of 360°.

Jai Ganesh · 2019-12-06 14:43:10

mikau wrote:

2520 is my favorite number. I found her one day when I decided to find the least common multiple of the numbers 1 through 10 and we've been friends ever since!
2 * 2 * 2 * 3 * 3 * 5 * 7 = 2520.
Not only is this number divisible by 1 through 10, it is divisible by any product of the above factors. Which produce the following:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72, 84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630, 840, 1260, 2520,
48 numbers total!
You can also multiply 2 * 2 * 2 * 2 * 3 * 3 * 5 * 7 * 11 * 13 * 17 * 19 to get 232,792,560, a number divisible by 1 through 20, and every product of these numbers. But this number is large and hard to remember, not nearly as sexy as 2520. Relatively small, and easy to remember! :-D

In a similar manner, the least common multiple of all natural numbers from 1 to 1000 would be roughly

.

Math Is Fun Forum

#1 2006-02-02 13:55:44

2520 A wonderfull little number

#2 2006-02-02 14:10:12

Re: 2520 A wonderfull little number

#3 2006-02-02 14:22:31

Re: 2520 A wonderfull little number

#4 2006-02-03 11:01:12

Re: 2520 A wonderfull little number

#5 2006-02-03 13:15:11

Re: 2520 A wonderfull little number

#6 2006-02-05 14:34:13

Re: 2520 A wonderfull little number

#7 2006-02-28 07:04:28

Re: 2520 A wonderfull little number

#8 2006-02-28 07:11:16

Re: 2520 A wonderfull little number

#9 2006-02-28 07:18:50

Re: 2520 A wonderfull little number

#10 2006-02-28 07:24:08

Re: 2520 A wonderfull little number

#11 2006-03-01 09:46:29

Re: 2520 A wonderfull little number

#12 2006-03-03 09:52:00

Re: 2520 A wonderfull little number

#13 2019-11-14 14:31:31

Re: 2520 A wonderfull little number

#14 2019-12-06 14:43:10

Re: 2520 A wonderfull little number

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