181 questions linked to/from How do I compute $a^b\,\bmod c$ by hand?
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### Fermats little theorem, $p$ is not a prime number [closed]

Calculate the remainder $$r \equiv 37^{877} \bmod{323}$$ I don't know how to follow this up since $323$ is not a prime number.
To determine this for $13^{33} \pmod{64}$ is easy since $\phi(64)=32$ and $\gcd(64,33)=1$, we have $13^{\phi(64)}=13^{32}\equiv1 \pmod{64}$. This means that $$13^{33}=13^{31}\cdot13\equiv 13 \equiv1 \... 4answers 65 views ### Check if an expression is divisible by 2016. (modulo operations?) How to check whether 29^{576} - 1 can be divided by 2016 without computing the numbers? I suppose that I have to use modular arithmetic, but don't really know how... 1answer 106 views ### How can I reduce this 3456^7 \cdot 3456^7 \pmod{2099}? [closed] The original form was 3456^{14} \pmod{2099}, but now I need to break down the 3456 so that I may perform the modulo operation 2answers 80 views ### Remainder when 3^{12} + 4^{21} is divided by 13. What is remainder when 3^{12} + 4^{21} Is divided by 13? I tried this question by found binomial theorem but as there are two bases it is becoming difficult to calculate . I am a school going ... 1answer 44 views ### How to find the last digit of a number in base b? For a number a^{x^{...^n}} . To find its last digit in a base b, Imagine that I have this number a^{x^{y}} to simplify the problem. Then I calculate a^{x} \equiv c \pmod b and after that c^{... 1answer 88 views ### calculate modular algorithm with large exponents [duplicate] I don't know how to calculate the following modulo:$$321^{654} \mod 1013$$Are there some easy way to do this? 0answers 85 views ### How to compute a big number modulo x? [duplicate] I still have problems computing (without a calculator) for example: R_{21}(3^{234}) I know i have to use the chinese reminder theorem, bu before that, how do i get there? I know that i haw to split ... 2answers 58 views ### Easy Number Theory modular exponentiation So I was reading through some old questions I found For the question I was asked to explain why for \,2018^{\large 2017^{\LARGE n}}\!\!\bmod 1001 Where n is an integer between 14 million and 17 ... 3answers 70 views ### What will be the remainder if 7^{101} mod 5? [closed]$$7^{101} \mod 5 = 49^{100} \mod 5 = (49^2)^{50} \mod 5$$What is the next step to find solution? Regards 4answers 34 views ### Large Remainders with Mod Determine the remainder of 5^{2017} when divided by 7. I know that we need to use mod 7 to find all of the different remainders but I am not sure what specific steps to take and how to finish it ... 2answers 33 views ### Congruence with powers How would one go about solving the following congruence: x\equiv2^{2012} \pmod6 I assume you can divide that number on the left to lower it, but how exactly? Same issue applies to x\equiv6^{29} \... 1answer 53 views ### Solving the equation for x I need help solving for x on the following congruence x^{49943} \equiv 10855 mod (63571). I started computing \phi (63571) = 63000, where \phi is the Euler Phi function. Next, we solve ... 2answers 36 views ### Modulus algorithm for finding a*b^c mod n, avoiding large numbers? I know the algorithm finding  (a^b) mod\;n  avoiding large numbers so I can code it, but I'm wondering if anyone can help me with a similar algorithm for$$ (a\cdot b^c )mod\;n  It's quite hard ...
I want to solve this modular equation: $7x + 5 \equiv 2^{11^{2017}} \pmod {31}$ As far as I know, dividing the exponent by 31 and substituting it with the remainder is not allowed. I've looked ...