# Tagged Questions

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

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### Proving $1+2^n+3^n+4^n$ is divisible by $10$

How can I prove $$1+2^n+3^n+4^n$$ is divisible by $10$ if $$n\neq 0,4,8,12,16.....$$
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### What does the “or” symbol mean as in “$d\mid a$”

What does the "or" symbol mean as in the following post: How to prove $\gcd(a,\gcd(b, c)) = \gcd(\gcd(a, b), c)$? In particular, the symbol is used in the above linked post in the following ...
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### Prove that $\sqrt{n^2 + 2}$ is irrational

Suppose $n$ is a natural number. Prove that $\sqrt{n^2 + 2}$ is irrational. From looking at the expression, it seems quite obvious to me that $\sqrt{n^2 + 2}$ will be irrational, since $n^2$ will be ...
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### Divisibility of numbers between $n^3$ and $n^3+n$

Let $n$ be a positive integer. Given are numbers $n^3,n^3+1,\ldots,n^3+n$. Of them, $a$ are colored red, and $b$ others blue. The sum of the red numbers divides the sum of the blue numbers. Prove that ...
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### Show natural numbers ordered by divisibility is a distributive lattice.

I need a proof that the set of natural numbers with the the relationship of divisibility form a distributive lattice with gcd as AND and lcm as OR. I know it can be shown that a AND (b OR c) >= (a ...
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### Is there a quick parity test for integers expressed with odd radicies?

For integers expressed with an odd base, is there an easy way to tell if the number is odd or even? For an even base, if the ones digit is even, so is the integer. But this doesn't hold true for odd ...
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### Alternate way to Prove or disprove $6\mid n(n+1)(n+2)$

This is my proof, I'm wondering if I'm correct, and how to do without induction. My Work Basis Step $$\frac{(1)(2)(3)}{6} = 1$$ Inductive Hypothesis Assume that $\dfrac{k(k+1)(k+2)}{6} = d$ where ...
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### Proving an identity of the Möbius function and Euler’s totient function product

Could anyone kindly help me to prove that $$\sum_{d|n} \mu(d) \varphi(d) = 0$$ for all even integers $n \geq 2$, where $\mu$ is the Möbius function and $\varphi$ is Euler’s totient function? ...
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### Prove that, $(2\cdot 4 \cdot 6 \cdot … \cdot 4000)-(1\cdot 3 \cdot 5 \cdot …\cdot 3999)$ is a multiple of $2001$

Prove that the difference between the product of the first 2000 even numbers and the first $2000$ odd numbers is a multiple of $2001$. Please show the method. I have started with the following ...
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### Application of the Jacobian

I have been stuck on this question for a while now to no success. Help would be appreciated. Consider $x$ and $y$ such that $(x, p) =(y, p) = 1$. For what $p$ does their exist $x$ and $y$ such ...
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### Modulo Arithmetic of Complex Numbers

Suppose $a,b,c \in \mathbb{C}$ such that $$a+b+c\in \mathbb{Z},$$ $$a^2+b^2+c^2=-3,$$ $$a^3+b^3+c^3=-46,$$ $$a^4+b^4+c^4=-123$$ then find $(a^{10}+b^{10}+c^{10})\pmod{1000}$. I only observed that ...
If we consider the integers modulo a prime $p$, then for every $x \not \equiv 0$ (mod $p$), we can get any $b \not \equiv 0$ by adding $x$ a number of times to itself. Is the same true for ...