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At first sight, this number will be considered a normal number. But the strange thing is 2520 is able to be divided at even or odd number. Like 1=2520, 2=1260 , 3=840, 4=630, 5=504, 6=420, 7=360, 8=315, 9=280, and by 10=252, which is hard to find integers with the same characteristics.

Also, you can get this number by (7 * 30 * 12 = 2520) which I think is 7 days in the week, 30 days in a month and 12 months in a year! Nothing special, but it is a weird thing, right.

Are there any numbers that can be divided from 1-10 without any fraction? If yes, what are they? And is there any mathematical explanation of such a phenomenon?

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    $\begingroup$ This phenomen is related to highly composite numbers. The number $2520$ is $lcm(2,3,\cdots ,9,10)$, so the smallest number divisibly by every number from $2$ to $9$ $\endgroup$
    – Peter
    Mar 8, 2017 at 14:37
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    $\begingroup$ $2520$ is the Least Common Multiple of the integers $(1,2,\cdots, 10)$. Any number divisible by that will work. $\endgroup$
    – lulu
    Mar 8, 2017 at 14:37
  • $\begingroup$ What exactly do you mean by "2520 is able to be divided at even or odd number"? $\endgroup$
    – Arthur
    Mar 8, 2017 at 14:43
  • $\begingroup$ @Arthur I think OP meant between 1 and 10. $\endgroup$ Mar 8, 2017 at 14:43
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    $\begingroup$ By the way, "$1=2520$" is a highly questionable statement... It's a false statement, actually. I know what you mean, but you must express your thoughts correctly. $\endgroup$
    – zipirovich
    Mar 8, 2017 at 16:13

5 Answers 5

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If you want a number divisible by $a, b,$ and $c$, you can just multiply them together. So one number that is divisible by $1$ through $10$, is the product $1\cdot 2\cdot \cdots \cdot 10 = 3628800.$ The smallest such number will be the least common multiple of $1, 2, \ldots, 10$ which is your $2520.$

Is it "weird" that $2520 = 7\cdot 30 \cdot 12$? Not really. Before metric ruined everything, we chose our units to be divisible by small numbers. There are lots of $12$'s and $30$'s (and $60$'s, etc) in our old measuring systems. And since this problem depends on having lots of small factors, it would be expected that the least common multiple of the first few numbers would contain the same factors as many of the ancient units.

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Every number can be factored as a product of primes (with exponents) in a unique way. For example,

$$2520=2^3\cdot3^2\cdot5\cdot 7.$$

If you want to know the smallest number that is a multiple of the numbers $1$ to $16$, factoring as

$$1,2,3,2^2,5,2\cdot3,7,2^3,3^2,2\cdot5,11,2^2\cdot3,13,2\cdot7,3\cdot5,2^4$$

you need to take all these primes with sufficient exponents (the largest), giving

$$2^4\cdot3^2\cdot5\cdot7\cdot11\cdot13=720720.$$


The consecutive "secret numbers" are

$$1,2,6,12,60,60,420,840,2520,2520,27720,27720,360360,360360,360360,720720,12252240,\cdots$$

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    $\begingroup$ Maybe you want to add a link to oeis.org/A003418 since you already included the first few terms of the sequence in your answer? $\endgroup$ Mar 8, 2017 at 22:12
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$2520$ is the least common multiple of the $10$ first natural numbers. Therefore, this is the least pòsitive integer that is a multiple of the numbers from $1$ to $10$. But there are infinitely many numbers with this property. Any multiple of $2520$ ($5040$, $7560$, $10080$, etc) has this property, too.

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$2520 = 2^3 \cdot 3^2 \cdot 5 \cdot 7$ i.e the primefactorisation of 2520. From the primes you see that it will be divided by all numbers $1-10$, since you with use of the primes can construct all the number $1-10$.

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To add to this one observation: refactor 7 * 30 * 12 = 2520 to 7 * 360 = 2520 (as of course 12 * 30 = 360). Your number here, 2520 is equal to 7 circles, in terms of degrees (that would bring up the question: is 360° just a convention or a true mathematical reality, I believe it is the latter, but won't defend that here). So that makes it a very special number. That should answer part of your question at least:

Also, you can get this number by (7 * 30 * 12 = 2520) which I think is 7 days in the week, 30 days in a month and 12 months in a year! You were already turning a year into 360 days (in effect at least), so if we were to allow that, it would just be a circular statement, 360 (days in this case) * 7 ('years' in this case) of course simply equates to 2520.

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  • $\begingroup$ Downvoters, care to be man enough to explain what is wrong with my point? It exactly answers the second of the issues addressed in the question. $\endgroup$ Mar 9, 2017 at 21:57
  • $\begingroup$ The second part of the answer asks for numbers other than $2530$ with similar interesting properties. Where are they in your answer? I don't see them. $\endgroup$
    – Alex M.
    Mar 16, 2017 at 17:08
  • $\begingroup$ The author addressed / questioned on multiple issues. I addressed one of them, and I said so at the beginning of my answer: "To add to this one observation," and I cited the specific part of his question I was answering. Perhaps you are being too inflexible. I was not posturing my post as the main answer to the comprehensive question. Also, I think the observation I made is fundamentally amazing, that this gets to the nature of a circle. I am the only one here who related this amazing number to geometric reality. $\endgroup$ Mar 16, 2017 at 17:46
  • $\begingroup$ You could improve your answer by explaining why 360 is the secret to this nice property of 2520, e.g. some history on why it's decided that there are 360 degrees in a complete circle. The part where you explain that OP fails to notice that a year is now left with only 360 days, should be more of a comment to the question, than an answer. $\endgroup$
    – Frenzy Li
    Jul 16, 2017 at 12:44

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