What is the secret of number '2520'?

Question

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?

Answer 1

$2520$ 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.

Answer 2

$2520 = 2^3 \cdot 3^2 \cdot 5 \cdot 7$ 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 $1-10$.

Answer 3

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.

Answer 4

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

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

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The consecutive "secret numbers" are

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

Answer 5

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.