What is the remainder when $2^{1990}$ is divided by $1990$?

Question

I actually do not have the basic idea on how to approach these type of questions....so please tell me a generalized method about all this too.

It came in RMO, and the question is:

What is the remainder when $2^{1990}$ is divided by $1990$ ?

Answer 1

Using Fermat's Little Theorem,

$2^4\equiv1\pmod 5\implies 2^{1990}=2^2\cdot(2^4)^{997}\equiv4\cdot1^{997}\pmod 5\equiv4\ \ \ \ (1)$

$2^{198}\equiv1\pmod {199}\implies 2^{1990}=2^{10}\cdot(2^{198})^{10}\equiv1024\cdot1^{10}\pmod{199}\equiv 29\ \ \ \ (2)$

Clearly, $2^{1990}\equiv0\pmod2\ \ \ \ (3)$

Apply the famous CRT on $(1),(2),(3)$

Answer 2

I was lazy to do the calculations required when applying the Chinese Remainder Theorem, so in the end I managed to come up with the following solution.

First note that $199$ is prime. By Fermat's Little Theorem we have, $$2^{198} \equiv 1\pmod{199} \implies 2^{199} \equiv2\pmod{199} \implies 2^{1990} \equiv 2^{10} \pmod{199}$$

$$ \implies\boxed{199\mid (2^{1990}-2^{10})}$$

We can also observe that $2^{1990}$ and $2^{10}$ have the same last digit, $4$, since $1990 \equiv 10 \equiv 2\pmod{4}$.

$$\implies\boxed{10 \mid (2^{1990}-2^{10})}$$

From the above we conclude that $$1990\mid (2^{1990}-2^{10}) \implies 2^{1990} \equiv 2^{10} \equiv 1024 \pmod{1990}$$

Therefore, the remainder when $2^{1990}$ is divided by $1990$ is $\boxed{1024}$

Answer 3

Let $ x = 2^{1990}$
$1990 = 2 \cdot 5 \cdot 199$

Find $x_2 = x \pmod 2$
$\qquad 2^{1990} \equiv 2\cdot2^{1989} \equiv 0 \pmod 2$
$\qquad x_2 = 0$

Find $x_5 = x \pmod 5$
$\qquad 2^4 \equiv 1 \pmod 5$
$\qquad 2^{1990} \equiv 2^{4\cdot 497 + 2} \equiv 2^2 \equiv 4 \pmod 5 $
$\qquad x_5 = 4$

Find $x_{199} = x \pmod{199}$
$\qquad 2^{198} \equiv 1 \pmod {199}$
$\qquad 2^{1990} \equiv 2^{198 \cdot 10 + 10} \equiv 2^{10} \equiv 1024 \pmod{199} \equiv 29 \pmod{199}$
$\qquad x_{199} = 29$

Factors of $1990$ are $2, 5, 199$

Find $\alpha_2$ such that $\alpha_2 \equiv (1,0,0) \bmod (2,5,199)$
$\qquad 5 \cdot 199 = 995$
$\qquad 995 \equiv (1,0,0) \bmod{(2,5,199)}$
$\qquad \alpha_2 = 995$

Find $\alpha_5$ such that $\alpha_5 \equiv (0,1,0) \bmod (2,5,199)$
$\qquad 2 \cdot 199 = 398$
$\qquad 398 \equiv (0,3,0) \bmod {(2,5,199)}$
$\qquad 3^{-1} \equiv 2 \pmod 5$
$\qquad 2 \cdot 398 = 796$
$\qquad 796 \equiv (0,1,0) \bmod {(2,5,199)}$
$\qquad \alpha_5 = 796$

Find $\alpha_{199}$ such that $\alpha_{199} \equiv (0,0,1) \bmod (2,5,199)$
$\qquad 2 \cdot 5 = 10$
$\qquad 10 \equiv (0,0,10) \bmod {(2,5,199)}$

$\qquad \qquad $ Find $10^{-1} \pmod{199}$ \begin{array}{r|rrr} & 199 & 1 & 0 \\ -20 & 10 & 0 & 1 \\ & -1 & 1 & -20 \\ & 1 & -1 & \color\red{20} \\ \end{array}
$\qquad \qquad 10^{-1} \equiv 20 \pmod{199}$

$\qquad 20 \cdot 10 = 200$
$\qquad 200 \equiv (0,0,1) \bmod {(2,5,199)}$
$\qquad \alpha_{199} = 200$

$x = \alpha_2 \, x_2 + \alpha_5 \, x_5 + \alpha_{199} \, x_{199} \pmod{1990}$
$x = 0 \cdot 995 + 4 \cdot 796 + 29 \cdot 200 \pmod{1990}$
$x = 1024$

Answer 4

As the $(2^{1990},1990)=2$

let us start with $\displaystyle\frac{2^{1990}}2\pmod {\frac{1990}2}$ i.e., $2^{1989}\pmod {995}$

Now $\displaystyle995=5\cdot 199$ and using Carmichael function, $\lambda(995)=396$

As $\displaystyle(2,995)=1, 2^{396}\equiv1\pmod{995}$

As $\displaystyle1989\equiv9\pmod{396},2^{1989}\equiv2^9\pmod{995}$

$\displaystyle\implies2^{1989}\equiv512\pmod{995}\ \ \ \ (1)$

Now we know if $x\equiv y\pmod m, a\cdot x\equiv a\cdot y\pmod {a\cdot m}$ where $a$ is any integer

So, multiply either sides of $(1)$ by $2$