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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Need help proving or disproving the following idea I had (basic number theory, modular arithmatic)
If a sum of n numbers has remainder 1 after division by n. Does there exist a sub-sum of those numbers that have remainder 1 after division by n.
Put more formally:
Given a sequence $(a_1, a_2,...,a_n)...
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Adding a number to a multiple of nine does preserve its digital root. [duplicate]
If we add any number to a multiple of nine, then the summation of the digits of the addent (till it comes to single digit)is equal to the summation of the digits of the result(till it comes to single ...
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Order of a polynomial $P\pmod{n}$
Let $P$ be a polynomial in $\mathbb{Z}[X]$ and $n \geq 1$ be an integer.
Consider the vector $\left(P(0), P(1),\ldots,P(n-1) \right) \pmod{n}$
Now apply $P$ again, and again, pointwise to the vector, ...
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Is there a term for [a - x(mod(a))]?
I found this code snippet in a code base, and the function is obviously named wrong:
const mod11 = (value: number): number => 11 - (value % 11);
But it is not &...
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Find the remainder (factorial with Wilson's theorem) [duplicate]
"Find the reminder of $(1001!\cdot 994!)^{19961} (\text{mod} 1997)$ if we know 1997ドル$ is a prime number."
Okay so that means:
$$x\equiv (1001!\cdot 994!)^{19961} (\text{mod} 1997)$$
we know ...
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Is the product of two $k$-th power non-residue a $k$-th power residue? [closed]
Let $p$ be a prime, then $a\in\{0,\dots, p-1\}$ is a $k$-th power residue modulo $p$ if there exists some integer $x$ such that $x^k\equiv a $(mod $p$) and a $k$-th power non-residue otherwise.
It is ...
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Stabilization of one new trailing digit of $a^{10^b}$ as $b$ increases, for every $a$ not divisible by 10ドル$
Let $a,b \in \mathbb{N}-\{0,1\}$ be such that 10ドル \nmid a$.
Then, define
$S_a(b) := \nu_{10}(a^{10^{b+1}} - a^{10^b})$ and
$D_a(b) := S_a(b) - S_a(b-1)$.
So, $S_a(b)$ counts the number of common ...
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Solving this equation for k, don't know where to begin [duplicate]
If $k$ and $N$ are both integers then is there an expression for the minimum of $k$ if $N$ is given and
10ドル^k\mod{N}=1$?
What I mean is, is there a function $f$ such that $f(N)=k$?
I haven't seen ...