# Questions tagged [modular-arithmetic]

Modular arithmetic (clock arithmetic) is a system of integer arithmetic based on the congruence relation $a \equiv b \pmod{n}$ which means that $n$ divides $a-b$.

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### Prove that the only solution to each of the following equations is (0,0)

Prove that the only solution to each of the following equations is (0,0): $n^2=5m^2$ $n^3=40m^3$ Thank you!
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### If n is not congruent to 2 modulo 4, then $n=a^2 - b^2$, with a, b integers.

This is the problem 5.54 in Childs - A concrete introduction to higher algebra. I found a proof of the inverse proposition, $n=a^2 - b^2$ then n is not congruent to 2 modulo 4,considering the ...
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### Proving any two elements in recursive sequence are coprime.

A sequence $x_0, x_1, x_2, \ldots$ is defined recursively as follows: $$x_0 = 3 \\ x_n = 2 + (x_0 \cdot x_1\cdot x_2\cdots x_{n-1})$$ I'm stuck at trying to prove that for any two different elements ...
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### $aRb$ if and only if b-a is divisible by both p and q. Prove that the set of all equivalence classes of R is equal to Zpq. [duplicate]

I am stuck with proving the following. I am given a relation 𝑎𝑅𝑏 a R b if and only if b-a (two integers) divide both p and q, with p and q being distinct primes. I have already proved that the ...
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### Counting solutions to $x^2+y^2 \equiv d \pmod{p}$ for a prime $p \equiv 3 \pmod 4$

For any given $p \equiv 3 \pmod{4}$ and $d=1, 2, \dots, p-1$, we would like to show that there are always exactly $p+1$ solutions to $x^2 + y^2 \equiv d \pmod{p}$. This conjecture comes from some ...
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### aRb if b-a is divisible by both p and q. Prove that the set of all equivalence classes is equal to $\mathbb{Z}_{pq}$

I'm stuck with proving the following. I am given a relation $aRb$ if and only if b-a (two integers) is divisible by both p and q, with p and q being distinct primes. I have already proved that the ...
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### Questions regarding the proof of the division algorithm

I have several questions regarding the following proof of the division algorithm. I added my question in brackets at the parts that I don't really understand. Thank you very much for your help! ...
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### Show that $2^{2^n} = (\prod {p_i^{a_i}}\equiv 2^{n+1}\alpha_ix_i+1) \mod 2^{2n+2}\implies 2^{n+1} (x_1 \alpha_1 + \dots + x_k\alpha_k )$

I had doubt in the following solution. However I couldn't understand the part " For this, it is enough to show that $x_i (\alpha_1 +\dots + \alpha_k ) \ge 2^{n+1}$" Then the author ...
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### Is this function, $f$, a homomorphism under addition and subtraction?

Allow me to construct $f$ $\textbf{Lemma}$: For all $x \in [0,1]$, there exists a sequence $\{b_k\}_{k=1}^\infty$ such that $$\forall k \in \mathbb{N}, b_k \in \{0,1\}$$ And x = \sum_{k=1}^\infty \...
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### Is $\{0,1,3\}$ a proper subgroup of $\mathbb Z _ 4$ under addition?

Is $\{0,1,3\}$ a proper subgroup of $\mathbb Z _ 4$ under addition? I think it is not because closure property does not hold for it. If we check, $3+3$ gives $2$ (in $\bmod 4$) which is not present ...
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### How to prove that $x + y \equiv 4\pmod 6$

Prove that if x and y are integers such that $x \equiv 3 \pmod{12}$ and $y \equiv 7\pmod{18}$, then $x + y \equiv 4\pmod6$ I tried making the equations into algebraic equations. So, $x\equiv3\pmod{12}$...
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