# Tagged Questions

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### Simplifying modulus expressions and an unknown expression? discrete math

I have a few questions below that I need help with a) I don't really understand what that symbol means and how to solve it b) How do u simplify this without a calculator c) I got 2^-r = 0, iss this ...
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### “Remainder” operation in mod 2^32

I debated posting this here, in the cryptography SE, or the programming SE. Obviously, I chose here, but I'm not confident in my choice... I'm attempting to "undo" a function, but I've hit a slight ...
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### Direct proof using modular arithmetic

Give a direct proof of $8\mid (3^n + 5^n)$ for all odd natural numbers. I know how to prove this by induction, I am not sure how to go about it using a direct proof. I would start by saying that ...
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### Compute $F_{1000} \bmod F_{11}$, where $F_n$ denote the Fibonacci numbers

Compute $F_{1000} \bmod F_{11}$, where $F_n$ denote the Fibonacci numbers. Progress: $F_{11}=89$ . I believe you should find the period of $F_n \bmod 89$ and use that to solve it. But I'm not not ...
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### $p>3$ prime, show that there exists $0<x,y<\sqrt{p}$ so that $p$ divides $cx-y$

Let $p>3$ be a prime number that does not divide $c$. Show that if $p>3$ there exists $x$ and $y$ with $0<x,y<\sqrt{p}$, such that p divides $cx-y$. I believe I've shown the above but for ...
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### Find the largest integer $n$ such that $10^n$ divides $10^6!$

Let $N=10^6!$ Find the largest integer $n$ such that $10^n$ divides $N$. Furthermore, compute the first digit and the last non-zero digit of $N$. I have some ideas that you should be able to use ...
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### Compute $F_{1000} \bmod 1001$, where $F_n$ denote the Fibonacci numbers

Compute $F_{1000} \bmod 1001$, where $F_n$ denote the Fibonacci numbers. I have tried using the fact that $F_{n^k} \bmod F_n = 0, k=1,2,3,...$ but that doesn't get me anywhere. Thanks!
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### How do you solve linear congruences with three variables.

Given \begin{cases} x+y+z &\equiv 1 \pmod{10} \\ x+2y+3z &\equiv 2 \pmod{10} \\ 2x+3y+6z &\equiv 3 \pmod{10} \end{cases} find $x,y,z$. How does one solve such a system of ...
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### The closed form of a sum of mod(k,m) where k goes from 1 to a arbitrary number.

Is there a closed form for $\sum_{n=0}^{C} mod(n,m)$ for arbitrary integers m ?
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### Proof related with prime numbers and congruence

How to (dis)prove this $(n-2)! \equiv 1 \mod n$ If n is said to be a prime number. I guess we'll have to use FERMAT’S LITTLE THEOREM, and I just don't know where to start from. Thanks in advance ...
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### Proofs related with odd numbers and modulo 8

In my problem I have $s! + s^{2P} \equiv 1 \mod 8$ where $s > 4, P \geq 1, s,P \in \mathbb{Z}^+$ I tried to follow that example's logic, but I could not get a result $n^2 \equiv 1 \mod 8$ ...
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### prime number related proof

I want to prove if following is true for every integer a,b and c $$a^2 - b^2 = cp$$ then p|(a+b) or p|(a-b) where p is a prime number. Any suggestion, help would be highly appreciated. Thanks ...
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### What is the identity for ab=2ab (mod 7)?

Using elements 1, 3, 5 write out a Cayley table. The operation for the table is ab = 2ab. For example 5*4= 5*4*2= 40 congruent to 5 (mod 7). What is the identity for this table?
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### Smallest integer x s.t. x! congruent to 0 (mod 216)

By guess and check I found x to be 9, but is there a more general way to solve this?
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### Find the smallest integer x s.t. x congruent to 1 (mod 1,2,3,4,5,6,7,8,9,10)

Don't really understand this question. If this is asking to find an x for each mod then the answer would be just be x+m...If this is asking to find an x that satisfies all mods, then this cant be ...
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### Modular arithmetic and one-to-one functions

Let $S = \{0, 1, 2, 3, · · · , 99\}$ . For each of the following functions $f : S \rightarrow S$ , determine whether it is one-to-one and onto, by computing its values for all $k ∈ S$: Function 1: ...