# What is meant by “evenly divisible”?

"What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?" Is it different from divisible?

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It is a school version of "divisible." Used to be fairly common. – André Nicolas Aug 19 '11 at 19:30
$\rm m\$ evenly divides $\rm\ n\$ means simply that $\rm\ n/m\$ is an integer. The "evenly" presumably means that the remainder upon division is $\:0\:.\:$ – Bill Dubuque Aug 19 '11 at 19:34
Subareas of mathematics have their own conventions. Is $-5$ a divisor of $20$? Probably one would be expected to answer yes. What is the sum of the divisors of $20$? Probably one would not be expected to say $0$. It sort of makes sense to qualify divisible, when one means that the quotient is an integer. After all, $5/20=0.25$. But the fact is that in mathematics beyond school mathematics, "evenly divisible" is uncommon. – André Nicolas Aug 19 '11 at 19:56
This is an interesting problem. Is the answer $\dfrac{20!}{10!}$? I discarded the numbers 1-10. If the number is evenly divisible by multiples, then it is divisible by the number. – mathguy80 Aug 20 '11 at 7:25
@mathguy80: The problem is just to find the least common multiple of $1, 2, \dots, 20$. Working it out, this is less than $\frac{20!}{10!}$. – chiaroscuro Aug 20 '11 at 10:23

Evenly divisible means that you have no remainder. So, 20 is evenly divisible by 5 since 20 / 5 = 4. Though, 21 is not evenly divisible by 5 since 21 / 5 = 4 R 1, or 4.2.

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evenly divisible = divisible .

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Evenly divisible is same as divisible. So, you are just looking for the L.C.M. of first $20$ natural numbers.

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Evenly divisible would mean that the number is divisible by any number completely. To answer your question, the correct answer is 20! (20 factorial).

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$20!$ is not the smallest such number. For example $19!$ already works, and that too is much bigger than needed. – Jonas Meyer Jan 25 '13 at 2:36