# Tagged Questions

This tag is for questions about divisibility, that is, determining when one thing is a multiple of another thing.

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### Find the remainder of $\sum_{i=0}^{99} 2^{i^2}$ when dividing by 7 and determine if the quotient is even or odd

I've recently had this problem in an exam and couldn't solve it. Find the remainder of the following sum when dividing by 7 and determine if the quotient is even or odd: $$\sum_{i=0}^{99} 2^{i^2}$$ ...
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### Problem to find all $n$ in following situation [closed]

Find all $n>1$ such that $1^{n} + 2^{n} + 3^{n} +\cdots + (n-1)^{n}$ divisible by $n$. I'm not good at Number Theory so , give elementary answer.
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### Total number of integral solutions to the factors of a given numbet

Let $a$ be a factor of $120$ then what are the total number of positive integral solutions to $xyz=a$ including 120. The answer is $320$ . After wasting almost $15$ mins in getting the factors of each ...
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### The product of n consecutive integers is divisible by n! (without using the properties of binomial coefficients)

How can we prove, without using the properties of binomial coefficients, the product of n consecutive integers is divisible by n factorial?
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### Factorial Divisibility

Let $a$ and $b$ be positive integers greater than one. With that in mind, $$(a \cdot b)!$$ is not necessarily divisible by: a) $$a!^b$$ b) $$b!^a$$ c) $$a! \cdot b!$$ d) $${2}^{ab}$$ By brute-...
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### Has there ever been an application of dividing by zero?

Regarding the expression $a/0$, according to Wikipedia: In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by $0$, gives $a$ (assuming $a\not= 0$), and ...
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### Show that among any consecutive $16$ natural numbers one is coprime to all others

Show that among any consecutive $16$ natural numbers one is coprime to all others. Is it useful to use the division algorithm on $16$? $16k,16k+1,16k+2,...16k+15$
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### How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$?

I can see that this works for any integer $n$, but I can't figure out why this works, or why the number $42$ has this property.
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### Is a function of $\mathbb N$ known producing only prime numbers?

It is well known that a polynomial $$f(n)=a_0+a_1n+a_2n^2+\cdots+a_kn^k$$ is composite for some number $n$. What about the function $f(n)=a^n+b$ ? Do positive integers $a$ and $b$ exists such ...
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### Any digit written $6k$ times forms a number divisible by $13$

Any digit written $6k$ times (like $111111$, $222222222222222222222222$, etc.) forms a number divisible by $13$. (source: a solution taken from careerbless) I tested with many numbers and it seems ...
I can't come up with a way of proving this: $$256 \mid 7^{2n} + 208n - 1\\ \forall n \in \Bbb N$$ I've tried by induction but couldn't see when to apply the inductive hypothesis... $$P(n+1) = 7^{2n+... 0answers 23 views ### How Euclidian Algorithm for division works with algebric expressions? I am attending an introductory Number Theory class for Computer Science focused on cryptography. I have done some exercises with integers number but I have two exercises in which appears algebric ... 1answer 24 views ### Finding quotient and remainder for a division We are starting with division and congruence in my algebra course... this is one of the first exercises for the division algorithm. I've done the first that were given with fixed values but now I have ... 3answers 1k views ### How to solve this algorithmic math olympiad problem? So, today we had a local contest in my state to find eligible people for the international math olympiad "IMO" ... I was stuck with this very interesting algorithmic problem: Let n be a natural ... 1answer 35 views ### Find all n \in \Bbb Z such that n^2 + n + 1 divide n^3-22 I need help with this problem: Find all n \in \Bbb Z such that n^2 + n + 1 divide n^3-22. I've got to a point where I know that n^2 + n + 1 | -21. So it should be among {{-21, -7, -3, -1, 1,... 1answer 82 views ### Prove that (a+b+c)^{333}-a^{333}-b^{333}-c^{333} is divisible by (a+b+c)^{3}-a^{3}-b^{3}-c^{3} Prove that$$(a+b+c)^{333}-a^{333}-b^{333}-c^{333}$$is divisible by$$(a+b+c)^{3}-a^{3}-b^{3}-c^{3}, where $a,,b,c -$ integers, such that $(a+b+c)^{3}-a^{3}-b^{3}-c^{3}\not =0$ My work ...
Suppose that $n$ is a sum of squares of three integers divisible by $3$. Prove that it is also a sum of squares of three integers not divisible by $3$. From the condition, \$n=(3a)^2+(3b)^2+(3c)^2=9(a^...