# Tagged Questions

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

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### Is there a Fermat-era proof of Theorem 69 from Dickson's Intro to NT?

In Dickson’s Introduction to the Theory of Numbers (Ch. VI, pp. 91-93), he gives the following [wonderful and wonderfully general] theorem. Theorem 69: All integral solutions of $$x^2-my^2=zw$$ ...
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### Why does $p \equiv 1,2,4 \pmod 7 \iff p \equiv 1,9,25 \pmod {28}$ where $p \equiv 1 \pmod 4$

Why does $p \equiv 1,2,4 \pmod 7 \iff p \equiv 1,9,25 \pmod {28}$ I can find primes and probably work this out but is there a quicker way? Edit: p is an odd prime and $p \equiv 1 \pmod 4$
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### Prove using induction on n that: $8\mid5^n+2(3^{n-1})+1$

How can we use induction to prove that $8\mid5^n+2(3^{n-1})+1$ for any natural $n$?
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### Proof about least prime numbers dividing n

Assume $n \in N$ is composite. Prove if p is the least prime number dividing n, then $p^2 \leq n$ Approach: I tried to write the first few prime and composite numbers but I didn't any patter. Any ...
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### Prove that if $n|5^n + 8^n$, then $13|n$ using induction

I have to prove using mathematical induction that if $n \ge 2$ and $n|5^n + 8^n$, then $13|n$. Please help me.
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### Proof that $3^n | 2^{3^n} + 1$

Question: Proof by induction that $3^n | 2^{3^n} + 1$. Attempt: $$2^{3^{n+1}} + 1 = 2^{3^n} 2^3 + 1 = 2^{3^n} 2^3 + 1 + 2^3 - 2^3 = 2^3( 2^{3^n} + 1 ) + 1 -2^3$$ And the first is $3^n |$ ...
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### Show that the equation $x^2+y^2+z^2=x^2y^2$ has no integer solution,except $x=y=z=0$

Show that the equation $x^2+y^2+z^2=x^2y^2$ has no integer solution,except $x=y=z=0.$ Let one of the $x,y,z$ be even number.Let $x=2p$ $x^2+y^2+z^2=x^2y^2$ This gives $y^2+z^2$ is also even,which ...
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### Find the value of $a$ if $x^2+y^2=axy$ has positive integer solution.

Find the value of $a$ if $x^2+y^2=axy$ has positive integer solution. My try: Let g.c.d of $x$ and $y$ is $d$ i.e.$(x,y)=d$ and let $x=dx',y=dy'.$ Then $x'^2+y'^2=ax'y'$ I am stuck here.The answer ...
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### Finding all triplets of positive integers with certain property

Fix a $k \in \mathbb{N}$ How do I find all sets of positive integers $a_{1},a_{2},...,a_{k}$ such that the sum of any triplet is divisible by each member of the triplet. I couldn't see ...
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### Solve $ord_x(2) = 20$

Given that the (multiplicative) order of $2$ mod $x$ is $20$, how can I work out what $x$ is?
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### Possible mis-interpretation in Project Euler #21

Here is the problem statement for Problem 21 of Project Euler. Let $d(n)$ be defined as the sum of proper divisors of $n$ (numbers less than $n$ which divide evenly into $n$). If $d(a) = b$ ...
### Modular Arithmetic: Computing last digit of $206746^{20}$ [duplicate]
I have been given the number: ${206746}^{20 }$and the problem wants me to compute the last digit using modular arithmetic. How would I go about this? I know that since the ones digit is 6, no matter ...
I am reading a math book. It states the following rule: For an integer $n$ greater than 1, let the prime factorization of $n$ be $$n=p^a_1p^b_2p^c_3...p^m_k$$ Here $a, b, c,..., m$ are ...