Question

Find the remainder when 32 power 33 34 is divided by 11

Answer 1

Answer:

32^(3334)

Step-by-step explanation:

Since we have to find the remainder when 32^(3334) divided by 11

we can solve it easily if we have some knowledge of congruence

When two elements are congruent.

If (a - b ) is divisible by n then a is congruent to b at mod n and we denote it as a = b ( mod n )

So

Since

32 - (-1) = 33

and

33 is divisible by 11

so by definition of congruence

32 = -1 (mod 11)

Now the congruence has another property that is we can take any power on both side but the congruence does not effected so we take the power 3334 on both sides and we get

32^(3334) = (-1)^3334 (mod 11)

On right hand side (-1) have even power so

(-1)^3334 = 1

So congruence becomes

32^(3334) = 1 (mod 11)

So by definition again

32^(3334) - 1 divisible by 11

This implies that

1 is remains as remainder when 32^(3334) is divisible by 11

So remainder is 1