# Tagged Questions

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### Prove for relatively prime numbers.

Prove that for relatively prime positive integers $a$ and $b$, the equation $ax+by=c$ must have non-negative integer solution if $c>ab-a-b$.
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### Solving Multiple Equations with Many Variables

Here's a problem I have stumbled upon, which may have a straightforward solution with linear algebra. If so, I cannot see it. Choose $n > 0 \in \mathbb N$, and consider the sequence of equations: ...
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### number of solution to the given equation.

a,b,c, are all non-negative integers such that a + b + c=100 and 1000a + 300b + 50c = 10000 How many such triplets are possible? i have tried to reduce ...
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### Matrix Multiplication Integer Solution

Given a matrix multiplication and a vector addition. (A,b has rational entries) $$Ax+b$$ how do i get an $x$ for that $Ax+b$ is integer or show that there is not such a solution? $x$ has no ...
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### Defining and expressing as a system of two equations. Is my answer good?

We wish to spend $\$164.00$by purchasing$10$books, some costing$\$15.00$ and other $\$17.00$. How many books of each price do we buy? My answer: let$x$= number of books costing$\$15.00$ and ...
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### What is the value of $a + b + c + d$ if the following equation holds?

If $a, b, c$ and $d$ are positive integers less than $7$ and $$a(7)^3 + b(7)^2 + c(7) + d = 901$$ What is the value of $a + b + c + d$? Is it related to consum of roots and product of roots?
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### Reminder of equation

I have a simple(maybe too simple) question - how to find the reminder of equation? For example: $(85^{74}+17^{95})^{15} \equiv \ ? \ (mod\ 13)$ I know that it is something simple, but I couldn't ...
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### Solve the comparison

I have difficulties with these type of problems: Solve the comparison: $\displaystyle67x + 17 \equiv 0\pmod{28}.$ I'm sure it is something very simple but I'm stuck on it more than $2$ hours :( . ...
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### Linear Diophantine equation in two variables with additional constraints

Given, $$aX + bY = c$$ where, $$c > b > a > 0;\quad X, Y > 0;\quad b\nmid c, a\nmid c$$ I want to find out if a solution exists as efficiently as possible (I'm not interested ...
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### Given $XX^\top=A$, solving for $X$

Not equal to this (my) own question. It's more general, probably more easy than the original question. All of the elements of $X$ and $A$ are integers. $XX^\top=A$ and $A$ is a symmetric matrix. ...
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### $XX^t=A$, $X=?$. Where $X \in \{0,1\}^{n \times m}$

The problem: $XX^t=A$, $\quad$ ($X_{ij}\in{0,1}$, $\quad$ $\sum_{j=1}^m x_{ij}=2$), $\quad$ $X=?$ Details: $n,m \in N$ $A \in \{0,1,2\}^{n \times n}$ $X \in \{0,1\}^{n \times m}$ $A$ is a ...
I have been trying to read a lot of literature concerning the above subject but I've not found anything useful to help my case. Suppose you're given a linear diophantine in $a_1,a_2,\ldots,a_k$ where ...
I have made up this "fun" problem. The first row of a 3x3 matrix is $(a_{11} a_{12} a_{13})$. The next row consists of the variables $h, g$ and $c$ in any order. The last row are distinct non-zero ...