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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
1 year, 11 months ago

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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. Prasun Biswas · 1 year, 11 months ago

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

But thanks. Learned something once again. Trevor Arashiro · 1 year, 11 months ago

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@Trevor Arashiro 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. Prasun Biswas · 1 year, 11 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 Kalpok Guha · 1 year, 11 months ago

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@Kalpok Guha Nice work @Trevor Arashiro and @Kalpok Guha . \(\ddot\smile\) I was waiting for so long for this set. Had fun solving all of them. Pranjal Jain · 1 year, 11 months ago

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@Pranjal Jain Thank you . Kalpok Guha · 1 year, 11 months ago

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