# Tagged Questions

Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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### New Criterion to Carmichael Numbers

Can anyone prove the following: If $p$ $=$ $a*b*c$ is a Carmichael Number, and the lcm (least common multiple) of ($a-1$), ($b-1$), and ($c-1$) is $m$, prove ($p-1$)/$m$ is prime. (This applies to ...
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### How to find the first integer making two progressions have gcd $> 1$

Is there a technique to efficiently find the first positive integer, $r$, that makes: $$\gcd(97+r, 106-r) > 1\text{?}$$
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### On the kernel of a certain module epimorphism $\mathbb{Z}^2 \to \mathbb{Z}/6\mathbb{Z}$

In order the construct a certain projective resolution of $\mathbb Z / 6 \mathbb Z$ I need to find the kernel of the ($\mathbb Z$-) module morphism: \epsilon_0 : \mathbb Z^2 \to \mathbb Z / 6 \...
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### Can the set of uneven number be written as the intersection of two sets?

Let $U = \{ n \in \mathbb{N} | n \equiv 1 (2) \}$. My question is: Can we find two proper subsets $M,N$ of $\mathbb{N}$ such that $M \neq N$ and $M,N$ are not equal to $U$ and $U = M \cap N$? It ...
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### Combinatorical problem [on hold]

$k$ is a natural constant.Determine $x,y,z$ knowing that $\binom{z+k}{x+y} + \binom{z}{x} \le k$ and $2x+y \le z$.
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### Comp Questions-Enumeration, Rates, Numbers, Geometry [on hold]

For each integer from 0 to 999, Michael wrote down the sum of its digits. What is the average of the numbers that Michael wrote down? It takes Jacob one and a half hours to paint the walls of a room ...
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### For which values of $n$ the sum $\sum_{k=1}^n k^2$ is a perfect square?

Question. For which values of $n$ the sum $\sum_{k=1}^n k^2$ is a perfect square? Clearly, $n=24$ is one such value, and I was wondering whether this is the only value for which the above holds. The ...
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### Prove that $p^2 - 4qr$ ($p,q,r$ odd natural numbers) is never a perfect square

The givens for the question: $p, q, r$ are odd natural numbers. We need to prove that $p^2 - 4qr$ is never a perfect square. Inspecting a few examples it seems to be true, but I have no idea where to ...
### Prove that ${2^n-1\choose k}$ and ${2^n-k\choose k}$ ar always odd. [duplicate]
How can I prove that ${2^n-1\choose k}$ and ${2^n-k\choose k}$ always returns odd numbers? It is possible to prove this by congruence? by the way : $0 \leq k \leq (2^n-1)$
Find all integer solutions to the following: $2x+10y-11z=1$ $x-6y+14z=2$ I am not quite sure how to do this... I know I will get equations in the end with each variable expressed in terms of ...