"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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11$\begingroup$ It is a school version of "divisible." Used to be fairly common. $\endgroup$– André NicolasCommented Aug 19, 2011 at 19:30
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3$\begingroup$ $\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 $\:0\:.\:$ $\endgroup$– Bill DubuqueCommented Aug 19, 2011 at 19:34
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1$\begingroup$ 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. $\endgroup$– André NicolasCommented Aug 19, 2011 at 19:56
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1$\begingroup$ @mathguy80: The problem is just to find the least common multiple of $1, 2, \dots, 20$. Working it out, this is less than $\frac{20!}{10!}$. $\endgroup$– JoshCommented Aug 20, 2011 at 10:23
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5$\begingroup$ Anyone came because of #ProjectEuler100 ? :D $\endgroup$– Anand UndaviaCommented Jan 8, 2020 at 4:03
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6 Answers
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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" 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.
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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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5$\begingroup$ $20!$ is not the smallest such number. For example $19!$ already works, and that too is much bigger than needed. $\endgroup$ Commented Jan 25, 2013 at 2:36
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Means the same as "divisible". Answer is $2\times3\times5\times7\times11\times13\times17\times19$.
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3$\begingroup$ ... which is not divisible by $16$, for example. $\endgroup$– JoffanCommented Apr 5, 2017 at 19:40