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# Collaboration set: Number Facts

In collaboration with Kalpok Guha, we made a set of problems based around the topic of "types of numbers". There are many types of numbers out there and here are just a few of them.

This set serves primarially to teach everyone about the complexity of numbers and the many interesting properties they have because math is $$\textit{beautiful}$$. Except for trig.... Trig is not pretty (lol, I'm bad at trig).

Here is a direct link to the set

here is another one because everyone knows two links are better than 1

Note: most of these problems are doable by hand using a NT approach but a CS approach works just as well.

The "special" numbers include:

Multiplicative perfect numbers- those whose factors when multiplied (including the number itself) yield $$n^2$$.

Amicable pair- a number$$n$$ whose factors sum to $$k$$ and the factors of $$k$$ sum to $$n$$.

Abundant number- a number whose factors including itself sum to greater than $$2n$$.

Deficient number- a number who's factors including itself sum to less than $$2n$$

Perfect number- a number who's factors including itself sum to $$2n$$

Narcissistic number- a number of $$s$$ digits that can be represented by the form $$\displaystyle \sum_{i=1}^s 10^{i-1}c_i$$ and can be written in the form $$\displaystyle \sum_{i=1}^s c_i^n$$ where $$c$$ is a whole number and for some positive integral value of $$n$$.

Or I'm simplier form, a narcissistic number is one that when each individual digit is summed to the nth power, their sum is the original number.

I must say making this problem set taught me a lot of cool properties about numbers, some that I can't post because I can't think of a problem for them.

I even learned how to spell narcissistic: n-a-r-c-i-s-s-i-s-t-i-c,

Note by Trevor Arashiro
2 years, 8 months ago

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2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

## Comments

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@Trevor Arashiro, what you defined in the section of "Narcissistic number" is actually the definition of Perfect Digital Invariant (PDI) and not of Narcissistic number. Note that for a PDI to be a narcissistic number, the power $$n$$ for the elements of the sum should be equal to the number of digits of the given narcissistic number.

Check out the Wikipedia link on Narcissistic number for details.

- 2 years, 8 months ago

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Yes, you would know what a narcissistic number is :3 lol

But thanks. Learned something once again.

- 2 years, 8 months ago

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I'm not a narcissist in the least, you know. I have almost no self-respect, let alone self admiration. :\ :3

And btw, I think you should edit the note to correct that part.

- 2 years, 8 months ago

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We have really worked hard on the problems.I have made a set collaboration set So please like and re-share

- 2 years, 8 months ago

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Nice work @Trevor Arashiro and @Kalpok Guha . $$\ddot\smile$$ I was waiting for so long for this set. Had fun solving all of them.

- 2 years, 8 months ago

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Thank you .

- 2 years, 8 months ago

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