# Tagged Questions

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

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### How do you evaluate the quadratic residue of 7 mod p?

How do you evaluate this quadratic residue? I've been playing around with some specific values and I suspect 1 if p is of the form 28k+/-1, 3, 9 and -1 if 28k+/- 5, 11, 13. I have no idea how to come ...
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### Find the smallest number which leaves remainder $8,12$ when divided by $28$ and $32$.

The question is- Find the smallest number which leaves remainder $8,12$ when divided by $28$ and $32$. My book gives directly a formula- Required number=Lcm(the two numbers;here 28 and ...
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### Show that $x^2 + y^2$ and $x^2 - y^2$ cannot both be perfect squares at the same time where $x, y \in \mathbb{Z}^+$.

Show that $x^2 + y^2$ and $x^2 - y^2$ cannot both be perfect squares at the same time where $x, y \in \mathbb{Z}^+$. I think that $x^2 + 2xy + y^2$ and $x^2 + y^2$ are not consecutive squares ...
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### Defining Logic Algebraically, Math Functions & Integers

Introduction I wanted to define some functions algebraically to be used as "logical conditions" that would be assigned to a term $t$ to "control" its value. Or in some other words, I wanted to ...
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### Problem on coprimality testing.

If we have integers $a,b,c$ with $gcd(a,b)=1$ and $0<a,b<c$ then for what $x,y$ with $0<a,b<x,y<c$ we $$\gcd(ac-bx,bc-ay)=1$$ hold and for what such $x,y$ we have ...
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### Finding a number $n$ and $k$ such that $nx+k$ will be a perfect square for any two given $x$.

Given two positive integers $x_1,x_2$, is it always possible to find positive integers $n$ and $k$ such that the expression $nx_i+k$ becomes a perfect square for each $i$ ?
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### Without using prime factorization, find a prime factor of $\frac{(3^{41} -1)}{2}$

Not sure how to go about this. Law of quadratic reciprocity and Euler's Criterion is recently learned material but I'm not sure how this applies.
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### Proof using deductive reasoning

I need to deductively prove that the sum of cubes of $3$ consecutive natural numbers is divisible by $9$. I can prove deductively that they are divisible by $3$ but so far any combination I choose ...
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### Is there anyway to find how many prime factors has a composite number without knowing them?

Let's call f(n) the function that gives us the number of different prime factors of a composite number n For example: f(24)=2 Let's call g(n) the function that gives us the number of prime factors of ...
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### Primes of the form $(2p)^{2}+1$, $p$ prime, have $h^{2}+1$ as a prime divisor?

I'm an undergraduate student and I usually ask questions here about things I'm struggling with in my academical mathematical studies, but this particular question is actually more like a curiosity. ...
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### divisibility theorem proof?

I have found in a book the proof for the divisibility problem that says: If $a$ and $b$ are integers and $b$ is not equal to zero, then there is a unique pair of integers $q$ and $r$ such that ...
### For which polynomials $f$ is the subset {$f(x):x∈ℤ$} of $ℤ$ closed under multiplication?
You surely know about the Brahmagupta–Fibonacci identity, $$(a_1^2 + b_1^2)(a_2^2 + b_2^2) = (a_1a_2 \pm b_1b_2)^2 + (a_1b_2 \mp a_2b_1)^2$$ which tells us that the product of two numbers, each of ...