Question

What will be the remainder when 47^37^27 is divided by 11.

Answer 1

Answer:

         5

Step-by-step explanation:

It is not clear if this is meant to be

     (47^37)^27 = $\displaystyle\bigl(47^{37}\bigr)^{27} = 47^{37\times 27}$

or

     47^(37^27) = $\displaystyle47^{37^{27}}$

I guess it is meant to be 47^(37^27), because this is more interesting.

First, consider the exponent 37²⁷ modulo 5.

37²⁷  ≡  2²⁷  =  2²⁴ × 2³  =  (2⁴)⁶ × 8  =  16⁶ × 8  ≡  1⁶ × 3  =  3  (mod 5).

So  37²⁷ = 5n + 3, for some integer n.

Now consider the given number modulo 11.

47^(37²⁷)  =  47⁵ⁿ⁺³  ≡  3⁵ⁿ⁺³  =  (3⁵)ⁿ × 3³  =  243ⁿ × 27  ≡  1ⁿ × 5  =  5 (mod 11).

So when divided by 11, the remainder is 5.