MHB Can you simplify a monstrous remainder problem using modular arithmetic?

  • Thread starter Thread starter Guest2
  • Start date Start date
  • Tags Tags
    Hard Remainder
AI Thread Summary
To calculate ${5^{2009}}^{1492}\mod{503}$, one can utilize modular arithmetic, specifically Euler's theorem and Fermat's little theorem. Since 503 is prime, the period of powers of 5 modulo 503 is 502, derived from 503 - 1. Simplifying the exponent involves finding ${2009 \cdot 1492} \mod{502}$ to reduce the complexity of the expression. Analyzing a smaller modulus, such as 7, can also help clarify the reasoning process. Understanding these concepts is crucial for tackling large exponentiation problems in modular arithmetic.
Guest2
Messages
192
Reaction score
0
Find ${5^{2009}}^{1492}\mod{503}.$

How do you calculate a beast like this?
 
Mathematics news on Phys.org
Do you know about Euler's theorem, or Fermat's little theorem? Powers of 5 are periodic modulo 503, so your expression can be simplified if you can find what that big exponent is modulo that period. Euler's theorem tells us that the period is divisible by divides 503 - 1 = 502 (since 503 is prime). Does that make sense?

If that doesn't help, what if you replaced 503 by, say, 7, does that make it simpler to reason about?
 
Last edited:

Thread 'Non-orthogonal bases'

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
View full post »
Thread 'Imaginary Pythagorus'

Thread 'Imaginary Pythagorus'

I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
View full post »

Similar threads

B Can modular arithmetic help us find remainders and unit digits?
Replies
1
Views
1K
MHB Can we use this method to efficiently check if $N$ is composite?
Replies
29
Views
5K
B Confused about ciphers in the book 'Arithmetic for the Practical Man'
Replies
13
Views
2K
Calculating Taylor Series Remainder: Finding an Upper Bound for n
Replies
4
Views
3K
Modular arithmetic with a variable modulus and fractions
Replies
1
Views
3K
Can you use modular arithmetic to solve this?
Replies
4
Views
2K
Lunatic Rantings about Modular Arithmetic.
Replies
3
Views
3K
Are non-integer moduli useful in modular arithmetic?
Replies
3
Views
4K
MHB Brute force calculation of P(200)
Replies
2
Views
10K
I Find Lowest Rung to Drop Bottles & Prevent Breakage - Problem Solving
Replies
26
Views
4K

Hot Threads

Recent Insights

Back
Top