trick question?

Question

what is remainder when you take 1999^1999 divided by 7,
plz explain thanks

Answer 1

because 7 is prime

1999^6 = 1 mod 7 (why : based on Fermats little Theorem)
so (1999^6)^333 = 1 mod 7
so1999^1998 = 1 mod 7

so 1999^1999 = 1999 mod 7 = 4

Answer 2

the remainder of 1999/7 is 4

Answer 3

1999^1999 = 1999^(2000–1) = (1999^2000)/4
When 1999 is divided by 7 remainder is 4 so as per remainder theorem when 1999^2000 when divided by 7 total remainder will be 4^2000
4^2000 = 4^5^400 = 1024^400 when 1024 divided by 7 remainder is 2 so when 1024^400 divided by 7 total remainder will be 2^400 = 2^10^40 = 1024^40 again when when divided by 7 total remainder will be 2^40
2^40 = 1024^4
So when 1024 divided by 7 remainde is 2 so total remainde is 2^4 = 16
We have taken 1999^2000
So total remainder = 16/4 = 4

Answer 4

First, 1999 = 4(mod 7)
So 1999^1999 = 4^1999(mod 7)
But 4^6 = 1(mod 7) by Fermat's little theorem
and 1999 = 6*338 + 1
So 4^1999 = (4^6)^338 * 4 = 4(mod 7).

Answer 5

no answer

i can't get the calculator to raise 1999 by 1999

but if u do have time raise 1999 by 1999 then divide by 7

then multiply 7 to the whole number answer you got

then subtract the 2nd answer from the 1st answer.

and that the remainder

Answer 6

1999 divided by 7 has remainder of 4 (1999 = 7*285 + 4).

1999^1999 = (7*285 + 4)^1999
=7N + 4^1999 (N is an integer)

4^1 divided by 7 has remainder of 4
4^2 divided by 7 has remainder of 2
4^3 divided by 7 has remainder of 1
4^4 divided by 7 has remainder of 4
And so on.

4^(n+1) divided by 7 has remainder 4 times the remainder of 4^n divided by 7.

If n divisible by 3, 4^n divided by 7 has remainder of 1.
If n divided by 3 has remainder of 1, 4^n divided by 7 has remainder of 4.
If n divided by 3 has remainder of 2, 4^n divided by 7 has remainder of 2.

1999 divided by 3 has remainder of 1 (1999 = 3*666 + 1)
4^1999 divided by 7 has remainder of 1

Let 4^1999=7M+1

1999^1999=7N + (7M+1)=7(M+N)+1
1999^1999 divided by 7 has remainder of 1

Answer 7

since a^a mod n=a mod n
let ur question--
1999^1999 mod 7=1999 mod 7=4
reply me ....

Answer 8

1999^1999 ??

can you tell me what is the meaning of this symbol (^)
ty

Answer 9

well ummmm you see its like. Wow i have no idea.