Linked Questions

-1
votes
1answer
122 views

Find the last four digits of $2^{27653 }- 1$ [duplicate]

Find the last four digits of $2^{ 27653} - 1. This is one of the questions in a mathematics contest. I have tried to find the sequence but I found it impossible. (I have used excel to do this and the ...
-2
votes
3answers
94 views

What is the remainder of $2^{4000}$ divided by 99? [duplicate]

Can someone guide me on how to find solution to such problems within a minute as that is the amount of time I will be given during my exams. also share what answer you get as I got different answers ...
0
votes
4answers
52 views

How do I find a remainder using congruence? [duplicate]

I'm asked to find the remainder of $305^{305}$ when it is divided by $42$. My progress: $305 \equiv 11 $ (mod 42) so $305^{305} -> 11^{305}$ $11^1 \equiv 11 $ (mod $42$) $11^2 \equiv 121 \equiv ...
0
votes
1answer
40 views

Find the remainder when $10^{20^{30}}$ is divided by $23$ [duplicate]

Find the remainder when $10^{20^{30}}$ is divided by $23$ I guess this question is fairly simple, but I just want to make sure I'm on the right track. My answer is shown below. If it is incorrect, ...
1
vote
1answer
95 views

What is the least residue of 49^4 modulo 23 [duplicate]

This is what I got so far but for some reason I feel like I am wrong. 49 is congruent to 26 mod(23). Therefore we have 49^4 is congruent to 26^4 = 456976 which is congruent to 456923 mod(23)but this ...
0
votes
3answers
90 views

Fermat's Little Theorem: $24^{(54n + 1)} +11 \equiv 1 \mod 19$, where $n \in \mathbb{Z}^+.$ [duplicate]

I'm supposed to use Fermat's Little Theorem $($ If $p$ is a prime and $p$ doesn't divide $a$, then $a^{p-1} \equiv1 \mod p)$ and to find the least residue of the following: $24^{(54n + 1)} +11 \...
1
vote
2answers
67 views

Finding the remainder of $823^{823}$ in the division by 11 [duplicate]

I wish to find the remainder $823^{823}$ in the division by $11$. I used the Euler-Fermat theorem that tells that: $a^{\phi(n)} \equiv 1$ $(\mod n)$, whenever $(a,n) =1$ First, Euler- Fermat tells ...
0
votes
4answers
110 views

Efficient way to solve $31^6 \mod 189$ without using any external devices. [duplicate]

Assuming basic knowledge of modulus arithmetic, is there a way, a trick or formulas that allow you to solve modulus question with very large numbers without using calculator? For example, $$76^7 \...
-2
votes
2answers
64 views

Find $700^{1734} \mod{347}$ [duplicate]

$$700^{1734} \mod{347}$$ I know how this could be calculated if I had $376$ in the exponent, using Fermat's theorem. But I have no idea how to approach this problem. What theorem's are appropriate to ...
1
vote
4answers
106 views

Why $1472222^{5555} + 14145555^{2228}$ is divisible by $7$? [duplicate]

Maybe someone can show why $1472222^{5555} + 14145555^{2228}$ is divisible by $7$? I just have written $$1472222^{5555} + 14145555^{2228}\equiv3^{5555}+4^{2228}\pmod 7$$ but what to do next? I don't ...
0
votes
2answers
47 views

modular exponentiation $14^{20} \pmod{33}$ [duplicate]

How do I find 14^20 mod 33 ? I tried writing 14^20 as 14^(2+4+8+6) but still no simplification. Should I just check all the power of 14 mod 33 and hope that some of them give nice numbers (1 or 2) ...
3
votes
1answer
44 views

If Euler Totient function fails other methods to find the remainder for the modular exponentiation [duplicate]

Modular exponentiation using Euler Totient Function for the below question. $$ 128 ^{343} \mod 527 $$ using totient function. Is there any other method to find the remainder of the question if ...
0
votes
1answer
38 views

What is $(2018^{2019} + 2019^{2018}) \pmod{7}$? [duplicate]

The solution should be within a page and please give full explanation. The original question is, What is $$\left(2018^{2019} + 2019^{2018}\right) \pmod{7} ?$$
-1
votes
1answer
61 views

$2^{947} (\bmod 1373)$ [duplicate]

I have that $2^{947} (\bmod 1373)$ how does one solve this without a calculator? Can you separate it into nice $2^x2^y$ or $(2^x)^y$? I'm really not sure how to go about this problem. Thanks for any ...
0
votes
0answers
60 views

Modulo for power [duplicate]

Question: calculate $8^{126} \pmod{9}$. My try: $$ 8^2 \pmod{9} = 1\\ 8^3 \pmod{9} = 8\\ 8^4 \pmod{9} = 1\\ .... $$ So for every even power the modulo is 1, otherwise it's 8. However, what is the ...

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