Find the least nonnegative residue
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Find the least nonnegative residue

Find the least nonnegative residue of $5^{18} \mod 11$. To do this I took $5^2 \equiv 3 \mod 11$. Then I did $(5^2)^5 \equiv 3^5 \mod 11$. And $3^5 \equiv...

Published: Aug 29, 2026

Determine whether the following statements are true or false, and then prove answers
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Determine whether the following statements are true or false, and then prove answers

Determine whether statement is true or false, and then prove answer. For all integers $a,b,c$: a) if $a|bc$ and gcd$(a,b) = 1$ then $a|c$ b) if $a|c$ and ...

Published: Aug 29, 2026

Finding the equation for a line tangent to a parametric curve
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Finding the equation for a line tangent to a parametric curve

I have the parametric equation $x = 2t - 1$ $y = 3t + 5$ $t = -1$ (defined as $t_0$) I am trying to find the line tangent to it. My book says if $x'(...

Published: Aug 29, 2026

Difference between horizontal and vertical line tests.
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Difference between horizontal and vertical line tests.

Trying to understand what the differnce is between a vertical and horizontal line test. If an equation fails the vertical line test, what does that tell y...

Published: Aug 29, 2026

Integrating $\sec^2 x$ from first principles
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Integrating $\sec^2 x$ from first principles

Is it possible to solve $\int \sec^2 x ~dx$ without knowing that $\frac{d}{dx}\tan x = \sec^2 x$?

Published: Aug 29, 2026

Preparations for reading Algebraic Number Theory by Serge Lang
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Preparations for reading Algebraic Number Theory by Serge Lang

I am eager to learn algebraic number theory. It seems that Serge Lang's Algebraic Number Theory is one of the standard introductory texts (correct me...

Published: Aug 29, 2026

How to prove the limit of a sequence using
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How to prove the limit of a sequence using "$\epsilon-N$"

I think I have a proper understanding of the general procedure, but I'm having difficulty manipulating my inequality so that I can isolate $n$ by its...

Published: Aug 29, 2026

Wolfram Alpha: How to show functions with abs() and sgn() as a piecewise function?
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Wolfram Alpha: How to show functions with abs() and sgn() as a piecewise function?

I was trying to obtain this result: integrate e^|x| dx The result is $$ \int e^{|x|} dx = \frac12 e^{-x} \left( (e^x-1)^2 \operatorname{sgn}(x) -2e^x +e^{...

Published: Aug 29, 2026

A nice number is an integer ending in 3 or 7 when written out in decimal. Prove that every nice number has a prime factor that is also a nice numbers.
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A nice number is an integer ending in 3 or 7 when written out in decimal. Prove that every nice number has a prime factor that is also a nice numbers.

My teacher just asked me a question like this but i do not know how to start and work it out at all. Can someone help me out with that?

Published: Aug 29, 2026

What is meant by
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What is meant by "maximal proper factors" of a integer?

I understand what is meant by proper factors, e.g. the proper factors of 36 are 2, 3, 4, 6, 9, 12, & 18. However I've just seen the phrase "maxim...

Published: Aug 29, 2026

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