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I was trying to solve a problem when I found this statement

As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24.

I understand what an abundant number is. My question is How can "24" be the smallest number that can be written as the sum of two abundant numbers?

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    $\begingroup$ $24 = 12 + 12$. The smallest number that can be written as the sum of two distinct abundant numbers is $30 = 12 + 18$. $\endgroup$ – Daniel Fischer Sep 2 '13 at 18:51
  • $\begingroup$ But 12 is just one number! I thought "the sum of (two) abundant numbers" that they should be different numbers!! $\endgroup$ – Kareem Sep 2 '13 at 18:53
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    $\begingroup$ It is understood as "can be written as $a + b$ where $a$ and $b$ are abundant, not necessarily distinct". $\endgroup$ – Daniel Fischer Sep 2 '13 at 18:54
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Suppose we have a non-empty set of natural numbers, which we'll call $X$ (in your case, $X$ consists of the abundant numbers). Then

  • the smallest natural number that can be written as a sum of two elements of $X$ is necessarily the two times the smallest element of $X$.

  • the smallest natural number that can be written as a sum of two distinct elements of $X$ is necessarily the smallest element of $X$ plus the next-smallest element of $X$ (assuming that $X$ does in fact have at least two elements).

In the case of $X=$ the abundant numbers, we therefore have that

  • the smallest natural number that can be written as a sum of two abundant numbers is $$12+12=24$$

  • the smallest natural number that can be written as a sum of two distinct abundant numbers is $$12+18=30$$

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  • $\begingroup$ I realized that 24 - 12 = 12 but actually I was confused as it says the sum of two abundant numbers and what came in my mind firstly is that they should be different numbers. Anyways thank you for your answer. $\endgroup$ – Kareem Sep 2 '13 at 18:58

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