Question

What is the remainder when 2000¹000 divided by 13 pick one option?

Answer 1

The remainder when $2000^1^0^0^0$ is divided by 13 is 3.

What is remainder?

The amount that is "left over" after performing a calculation is referred to as the remainder in mathematics. The integer that remains after multiplying one integer by another to create an integer quotient is known as the remainder in mathematics.

We have to find the remainder when $2000^1^0^0^0$  is divided by 13 .

Now clearly,

2000 = (83 x 13) + 11

So the remainder when 2000  is divided by 13  is 11 that is,

2000 ≡ 11( mod 13)

Then,

$2000^1^0^0^0^$ ≡  $11^1^0^0^0$(mod 13)             ..(1)

Now since 13  is a prime number and 13  does not divide 11 then from Fermat's little theorem we have,

$11^1^3^-^1$ ≡   1 mod 13)

$11^1^2$ ≡ 1( mod 13)

$11^1^2^*^8^3$ ≡  $1^8^3$( mod 13)

$11^1^2^*^8^3*11^4$ ≡ $11^4$ (mod 13)                ..(2)

Now,

1000 = (83 x 12) + 4

So,

$11^1^0^0^0$ ≡ $11^(^8^3^*^1^2^)^+^4*11^4$ (mod 13)

$11^1^0^0^0$ ≡ $11^4$ (mod 13)                ..(3)       by congruence (2)

Again,

121  ≡ 4 (mod 13)

$11^2$ ≡ 4(mod 13)

$(11^2)^2$ ≡ $4^2$ (mod 13)

$11^4$ ≡ 16 (mod 13)

$11^4$ ≡ 3 (mod 13)

So, from congruence (3) we get,

$11^1^0^0^0$ ≡ 3(mod 13)

Then,

$2000^1^0^0^0$ ≡ $11^1^0^0^0$ (mod 13)

$2000^1^0^0^0$ ≡ 3 (mod 13)

Hence, the remainder when $2000^1^0^0^0$ is divided by 13 is 3.

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