Frequent Questions
188,097 questions
107
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9
answers
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Congruence Arithmetic Laws, e.g. in divisibility by $7$ test
I have seen other criteria for divisibility by $7$. Criterion described below present in the book Handbook of Mathematics for IN Bronshtein (p. $323$) is interesting, but could not prove it.
Let $n = (...
- 5,643
14
votes
3
answers
10k
views
Mod of numbers with large exponents [modular order reduction]
I've read about Fermat's little theorem and generally how congruence works. But I can't figure out how to work out these two:
$13^{100} \bmod 7$
$7^{100} \bmod 13$
I've also heard of the Congruence ...
- 291
457
votes
24
answers
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How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?
How can I evaluate
$$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$?
I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
- 4,803
141
votes
12
answers
29k
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Modular exponentiation by hand ($a^b\bmod c$)
How do I efficiently compute $a^b\bmod c$:
When $b$ is huge, for instance $5^{844325}\bmod 21$?
When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for ...
- 50.8k
209
votes
8
answers
109k
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How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series:
$$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{...
- 22.9k
566
votes
31
answers
251k
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How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
How can one prove the statement
$$\lim_{x\to 0}\frac{\sin x}x=1$$
without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.
This is homework. In my math ...
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1331
votes
27
answers
156k
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Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because ...
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408
votes
36
answers
155k
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If $AB = I$ then $BA = I$
If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.
I do not understand anything more than the following.
Elementary row operations.
Linear ...
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201
votes
14
answers
21k
views
Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$?
I know there must be something unmathematical in the following but I don't know where it is:
\begin{align}
\sqrt{-1} &= i \\\\\
\frac1{\sqrt{-1}} &= \frac1i \\\\
\frac{\sqrt1}{\sqrt{-1}} &...
- 2,213
24
votes
5
answers
11k
views
Solving linear congruences by hand: modular fractions and inverses
When I am faced with a simple linear congruence such as
$$9x \equiv 7 \pmod{13}$$
and I am working without any calculating aid handy, I tend to do something like the following:
"Notice" that adding $...
- 19.9k
47
votes
8
answers
26k
views
Why $\gcd(b,qb+r)=\gcd(b,r),\,$ so $\,\gcd(b,a) = \gcd(b,a\bmod b)$
Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so?
Addendum by LePressentiment on 11/29/2013: (in the interest of http://meta.math....
- 1,701
885
votes
60
answers
158k
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Different ways to prove $\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$ (the Basel problem)
As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem)
$$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$
However, Euler was Euler ...
183
votes
12
answers
58k
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Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
For all $a, m, n \in \mathbb{Z}^+$,
$$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
Juan Liner
132
votes
3
answers
11k
views
Are all limits solvable without L'Hôpital Rule or Series Expansion
Is it always possible to find the limit of a function without using L'Hôpital Rule or Series Expansion?
For example,
$$\lim_{x\to0}\frac{\tan x-x}{x^3}$$
$$\lim_{x\to0}\frac{\sin x-x}{x^3}$$
$$\...
- 280k
87
votes
5
answers
37k
views
How to use the Extended Euclidean Algorithm manually?
I've only found a recursive algorithm of the extended Euclidean algorithm. I'd like to know how to use it by hand. Any idea?
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