Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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.

share|cite|improve this answer

evenly divisible = divisible .

share|cite|improve this answer

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

share|cite|improve this answer

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

share|cite|improve this answer
$20!$ 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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.