Tagged Questions

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 $b-a$.

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No square has a decimal expansion ending in 79

Show that no square number has a decimal ending in 79. More generally, find all possible two-digit endings for squares. Let any digit number ending at 79 be represented as $$a_nx^n+.....+7x+9$$ Plug ...
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How to apply Chinese Reminder Theorem to this congruence system?

\begin{align*} 17x & \equiv -15 \pmod{5}\\ -11x & \equiv 5 \pmod{3}\\ 23x & \equiv 15 \pmod{7} \end{align*} $5$, $3$, $7$ are coprime, so the system has solution mod $105$. I'm not sure ...
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Proof of the Ribet's theorem

My question is very simple : My goal is to read a proof the proof of the epsilon conjecture proven by Ken Ribet (1986) which is an ingredient of the proof of the Fermat Last Theorem (I want the ...
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Remainder modulo matrices.

Is it possible to have a consistent definition of $A\bmod_{left} B$ that respects matrix multiplication from left where $A$ and $B$ are two square matrices with non-negative entries? Is there a ...
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What are good parameters for an $ax+b \pmod{2^L}$ hash with distinct first n bits of the first $2^n$ inputs?

I'm hashing 64 bit integers via $ax+b \pmod{2^{64}}$. Good parameters mean that, given an $1 \leq n \leq 64$, the first $n$ bits of the first $2^n$ inputs are distinct. How should I chose $a$ and $b$ ...
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Calculating the last digit of exponent

I need to calculate the last digit of $723^n$.(For every positive integer $n$). If it was to calculate the last digit of $a^b$ when I know the value of $a$ and $b$,then it was easy- for example,If I ...
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Why k should be odd? [duplicate]

My teacher once said, for any positive number $\ n,$ $\ n^k - 1$ would always have $\ n-1$ as a factor for all positive odd values of $\ k$. Could anyone tell me the proof? I have written my ...
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Finding solutions in modulo

If I know $x$ modulo m and n, then under what conditions on m,n and p will i necessarily know $x$ modulo p? My initial guess is only in trivial cases, i.e. p is a multiple of m or n, but i cant seem ...
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Pattern in numbers

I bumped into this mathematical calculation while driving my car. With lot of traffic jams and lot of time to kill, I did some scrambling of the registration numbers of the cars in front of me. I ...
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How to approximate

I was reading a book and saw this approximation $(1 - 10^{-3})^{1023} \approx 2^{-1.476}$ I am wondering how it is calculated.
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How many more legs than seats are in the leftover inventory (use modular-arithmetic)?

I have difficult with this problem, and appreciate any help. The Seats R Us factory produces chairs with four legs and stools with three legs. The seats and legs are the same for both chairs and ...
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Find two integers between 1 and 100

Can anyone help me with this? Thank you very much! Problem: Find two integers between 1 and 100 such that for each: a) if you divide by 4, the remainder is 3; b) if you divide by 3, the remainder ...
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modular alzebra

Like alzebra can I write the following? 1.(a+b+c+d+. . .n) % m = (a%m) + (b%m) + . . . +(n%m). 2.(abcd. . .n) % m = (a%m) * (b%m) * . . . *(n%m).
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Basic question with coprimes and modulos

I started reading about Modular Arithmetic and solving some random basic exercises, and this one appeared: "Find an integer number $a$ such that any $b$ coprime with 34 is congruent to $a^k \mod34$ ...
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Binomial coefficients mod p

I want to find the following sum mod $p$ (prime number): Let $i\geq \frac{p-1}{2}$, $\sum_{k=i}^{p-1} \binom{k}{i}\binom{k}{p-i-1} \pmod{p}$ OK, I succeeded in simplyfying this argument to the ...
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Find the remainder when $11^{12}$ is divided by $13$. [duplicate]

I am looking for an easier way than mine to solve the problem. Problem: Find the remainder when $11^{12}$ is divided by $13$. Here is what I did. I simplify $11^{12}$ mod $13$ = $(–2)^{12}$ mod ...