# Tagged Questions

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### Proving that any common multiplication of two numbers is a multiplication of their least common multiplication

Im trying to prove that if there are to numbers $n,m$ (natural numbers), and their smallest common multipe is $k$, so that $k = n·i$ and $k = m·j$ for some $i,j$ natural numbers, any common multiple ...
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### What is the efficient way to calculate number of divisors of N that are divisible by 2?. [closed]

For example if a number is given let say 8 then its factors are 1,2,4,8 hence total numbers of divisors which are divisible by 2 are (2,4,8) that is 3.
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### The number of distinct multiples of composites greater than $n$ that can be factored into two naturals less than or equal to $n$

Given a list of composites between $n$ and $\lfloor \frac{n^2}{2} \rfloor$: What would be the most efficient way to count, for each composite, the number of its distinct multiples that can be ...
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### How to find a certain uppper bound (see details)?

What would be the most efficient way to find this upper bound? Given natural number n and a natural number d < n, find the ...
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### Show that $S=\frac{1}{a_1}+\cdots+\frac{1}{a_n}$ is not an integer.

So I have a bunch of integers $a_1,...,a_n$ and a prime number $p$ that divides only one of the numbers in this sequence, say $a_k$ I want to show, that $S=\dfrac{1}{a_1}+\cdots+\dfrac{1}{a_n}$ is ...
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### Largest K-multiple free set out of a fully ordered set

i'm struggling conceptually with this problem, i don´t know how to approach it in a clever way (without a computer, or at least without a brilliant algorithm). Mathematicians defined a k-multiple set ...
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### If $7$ is the first digit of $2^n$, what is the first digit of $5^n$?

Let $2^n = 7\cdot 10^x + p$ and $5^n = a\cdot 10^y + r$ And now what? (We're in base $10$)
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### Fact About Equality Proven by Euler in 1748 (Context: Integer Partitions)

I'm currently reading Integer Partitions by Andrews and Eriksson. In the introductory chapter (p.2), there is the following statement: ... The table would have a more efficient design: ...
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### Infiniteness of non-twin primes.

Well, we all know the twin prime conjecture. There are infinitely many primes $p$, such that $p+2$ is also prime. Well, I actually got asked in a discrete mathematics course, to prove that there are ...
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### Solve $x^2+x+3=0$ mod $27$

I was preparing for my Number Theory class for next semester and one of the questions that I came upon is to solve $x^2+x+3=0$ mod $27$. I have seen modular arithmetic before but never one that ...
El Gamal encryption involves picking $(p,g,b)$ which is our public key. We compute $b=a^x$ $mod$ $p$. Here, $x$ is the private key which we don't know. What are some efficient and strong algorithms ...
Discrete math practice problems prove whether true or false If $a^2$ divides $b^2$, then $a^3$ divides $b^4$ I think it is false because it is true that if $a^2$ divides $b^2$ then $a$ divides $b$ ...