## goformit100 2 years ago Find the remainder when 2^1990 is divided by 1990.

1. goformit100

@mayankdevnani

2. ParthKohli

Mod arithmetic :') @terenzreignz

3. terenzreignz

Why me? :/

4. ParthKohli

Because you.

5. goformit100

In this question How to square to so much power ?

6. terenzreignz

Might have to resort to totients..... @ParthKohli ?

7. goformit100

Sir if i use the exponent rule will it work here ?

8. ParthKohli

Ah! Euler's Theorem!

9. satellite73

factor 1990 first

10. terenzreignz

Time to doodle... $\large 2^{1990}=4^{995}$

11. goformit100

How to factor it ?

12. satellite73

how to factor 1990?

13. terenzreignz

What is 4^5? $\Large = 1024^{199}$

14. goformit100

Yes @satellite73

15. satellite73

try $$2\times 5\times 199$$

16. mayankdevnani

1990 = 10*199 = 2* 5* 199

17. terenzreignz

$\large 1024^{199}=1024\cdot 1024^{198}=1024\cdot 2048^{99}$

18. goformit100

try $$2\times 5\times 199$$ means ?

19. terenzreignz

Now let's start working some "mod magic" and reduce the bases at mod 1990 $\Large =_{(mod \ 1990)} \ \ 1024\cdot 58^{99}$

20. mayankdevnani

|dw:1367281405845:dw|

21. mayankdevnani

ok @goformit100

22. goformit100

ok So LCM can be take as a good way of factorizing numbers ok ?

23. terenzreignz

$\Large \equiv 1024\cdot 58 \cdot 58^{98}$

24. mayankdevnani

yaaa @goformit100

25. mayankdevnani

it is the best way!!!

26. terenzreignz

$\Large \equiv 1682 \cdot 1374^{49}$ gah... possibly inefficient...

27. goformit100

ok

28. terenzreignz

$\Large\equiv 678\cdot (-616)^{48}$

29. terenzreignz

This is daunting.

30. satellite73

$$1990=2\times 5\times 199$$ and $$2^2\equiv 4(5)$$ also $$2^{198}\equiv 1(199)$$ by fermat

31. goformit100

Mods Sir(s) I have to make you know that the Equation you are posting have not opened yet

32. satellite73

actually this is kind of a pain isn't it

33. amistre64

if you are seeing "math processing error", try refreshing

34. goformit100

What to do ?

35. terenzreignz

$\Large \equiv 678\cdot (1356)^{24}$

36. goformit100

Yo It's done...Mods you'll great REFRESHING WORKED. No SEE

37. goformit100

Now I can see

38. terenzreignz

$\Large \equiv 678 \cdot (-634)^{24}\equiv678\cdot (634)^{24}$

39. goformit100

2^1990 is divided by 1990 what to actually for this ?

40. amistre64

2^1990 2^11 = 2048 = 38 mod 1990 2^(11(180)+10) is what i had in mond :)

41. terenzreignz

I lack creativity, guys :)

42. goformit100

@mikaela19900630 you may too se the question I have posted now.

43. goformit100

Thank you all of you. I can do these type of question Now :)

44. goformit100

*from Now

45. terenzreignz

$\Large \equiv 678\cdot (-24)^{24}\equiv 678\cdot 24^{24}$

46. terenzreignz

cr*p... sorry $\Large \equiv 678\cdot (-24)^{\color{red}{12}}\equiv 678\cdot 24^{\color{red}{12}}$

47. goformit100

Thank You Very Much.

48. goformit100