Linked Questions

0
votes
4answers
44 views

how to calculate remainder of large numbers? (no calculator)

How do I calculate the remainder of $30^{29} \pmod {51}$? I cant use Fermat's little theorem since $51$ is not a prime number.
0
votes
0answers
25 views

Mod Arithmetic, involving multiplication. [duplicate]

$(12\cdot 53 ) \bmod 32$ In the above case, if I multiple 12 by 53 and then take mod then what is the use of modular arithmetic. The other way is: $(12\cdot 53 ) mod 32 = (12 mod 32 = 12) + (53 mod ...
2
votes
4answers
137 views

Evaluate $6^{433} \pmod {21}$ and a proving question

Question 1: Denote $a \mod b$ as $a \% b$, where $a$ and $b$ are some integers Evaluate 12^32475 % 21 The following is what I tried: ...
1
vote
3answers
37 views

If $n\geq 3$, $9^n \equiv a (\mod 100)$ and $9^{n+1} \equiv b (\mod 100)$, then $a+b=90$.

I noticed a pattern in the powers of 9 modulo 100. $9^1 \equiv 9 \pmod{100}$ $9^2 \equiv 81 \pmod{100}$ $9^3 \equiv 29 \pmod{100}$ $9^4 \equiv 61 \pmod{100}$ . . . and conjectured the following: ...
-1
votes
2answers
41 views

What is the unit digit? [duplicate]

Find the unit digit of a) $[(317^{24})^{18} + (713^{18})^{24} ]$ b) $[(243^{15})^{56} + (342^{56})^{15} ] $ I got my answers a) 2 and b) 7 is this correct?
4
votes
6answers
145 views

Find the units digit of $572^{42}$

The idea of this exercise is that you use the modulus to get the right answer. What I did was: $$572\equiv 2\pmod {10} \\ 572^2 \equiv 2^2 \equiv 4\pmod{10} \\ 572^3 \equiv 2^3 \equiv 8\pmod{10} \\ ...
5
votes
2answers
100 views

Find the last digit of $22^{23^{24^{25^{26^{27^{28^{\cdot^{\cdot^{\cdot^{67}}}}}}}}}}$

Find the last digit of $22^{23^{24^{25^{26^{27^{28^{\cdot^{\cdot^{\cdot^{67}}}}}}}}}}$ in base-$10.$ Just to clarify, I want to find the last digit of the power tower of consecutive numbers starting ...
3
votes
6answers
97 views

Find the remainder of $\frac{x^{2015}-x^{2014}}{(x-1)^3}$

Find the remainder of $\frac{x^{2015}-x^{2014}}{(x-1)^3}$. Let $P(x)=x^{2015}-x^{2014}=Q(x)(x-1)^3+ax^2+bx+c.$ If we put $x=1$ in $P(x)$ and $P'(x)$, we get $a+b+c=0$ and $2a+b=1$. Then: $c=a-1$. The ...
1
vote
2answers
94 views

Help on residue: $3^x + 22^y \equiv 15^z \equiv 15 \pmod{40} $

I am reading this note (click here and go to page 1994 for detail, in the proof of lemma 8), and found- $$3^x + 22^y \equiv 15^z \equiv 15 \pmod{40} $$ now, I can derive $3^x + 22^y \equiv 15^z \ \...
2
votes
3answers
224 views

What's the remainder when this huge number is divided by 45?

One of my friends recently gave a mock test of a math exam in which he was asked this horrific question. He asked the same to me and I was totally blank on looking at it. So, it will be a huge help if ...
4
votes
5answers
125 views

Find polynomial with some conditions

I need to find $f(x)\in \mathbb{Q}[x]$, such that: $f(x)\equiv 1 \pmod{(x-1)^2}$ $f(x)\equiv x \pmod{x^2}$ $\deg(f(x))<4$ So, what I understand so far is that: $(x-1)^2\mid f(x)-1$ $x^2\mid ...
2
votes
3answers
76 views

Theorem 57 An Introduction to the Theory of Numbers

The Theorem state: If $(k,m)=d,$ then the congruence $$(1)\ kx≡l(mod\ m)$$ is soluble if and only if $d|l.$ It has then just d solutions. In particular, if $(k,m)=1,$ the congruence has always just ...
2
votes
2answers
69 views

How to find Residue of Power Integer

I found congruence like below in various note- $$ 6^x \equiv 16 \pmod{20}$$ $$ 5^z\equiv 5 \pmod{20}$$ For any $z,x$ (perhaps, I didn't see any other condition). How residues $16, 5$ are found? ...
1
vote
4answers
112 views

mod cancellation: compute $\, n/2\bmod 6\, $ from $\,n \bmod m\,$ for even $n$

I am using the C++ language. I want to calculate these 2 expressions:- In our case, $x= 100000000000000000$ Expression(1) $$((3^x-1)/2)\mod7$$ The numerator $3^x-1$ is always divisible by $2$(...
0
votes
3answers
38 views

Exponentiation with odd number in modular arithmetic

Show that if $n\geq3$ is odd, then $2^n-1\equiv7\mod24$. I tried solving this backwardly. We want to prove that $2^3(2^{n-3}-1)=2^n-2^3\equiv0\mod24$. Since $\frac{24}{2^3}=3$, this leaves us to ...

15 30 50 per page