# 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?

• 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
• @mathguy80: The problem is just to find the least common multiple of $1, 2, \dots, 20$. Working it out, this is less than $\frac{20!}{10!}$. – Josh Chen Aug 20 '11 at 10:23
• Anyone came because of #ProjectEuler100 ? :D – Anand Undavia Jan 8 '20 at 4:03

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.

evenly divisible = divisible .

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

"Evenly divisible" is equivalent to the term divides.''

For example, we can say that "20 is evenly divisible by 5", which is equivalent to saying that "5 divides 20"; or in mathematical notation, $$5 | 20$$.

I am sure that the author of the question that you cite, "What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?" chose to use the term "evenly divisible" instead of the more ordinary "divides", because to opt for the latter one would need to say something long-winded like, "1|n" and $$2|n$$ and \ldots and $$20|n$$, where $$n$$ is the smallest such integer in question.

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

• $20!$ 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

Means the same as "divisible". Answer is $2\times3\times5\times7\times11\times13\times17\times19$.

• ... which is not divisible by $16$, for example. – Joffan Apr 5 '17 at 19:40