Question

Calculate ${\mathbf{2}}^{\mathbf{125}}\mathbf{mod}\mathbf{127}$using any method you choose. (Hint: 127 is prime.)

Short answer

$2^{125}\mathrm{mod}127$is equal to

Step-by-step solution

Given condition

To find the modulo of any large number we can divide that number into small parts and find the modulo of the smaller part and recognized the pattern to get the answer of the large integer.

Calculate 2125mod127

For,$2^{125}\mathrm{mod}127$, we can write$2^{125}$as

$2^{125}=2^{7\left(17\right)+6}$

Where$2^7\mathrm{mod}127$ can be written as,

$$\begin{gathered} 2^7\mathrm{mod}127=128\mathrm{mod}127 \\ =1 \end{gathered}$$

Then, for $2^{125}\mathrm{mod}127$, we have,

$$\begin{gathered} 2^{125}\mathrm{mod}127=2^{7\left(17\right)+6}\mathrm{mod}127 \\ =2^{7\left(17\right)}.2^6\mathrm{mod}127 \\ =1^{17}.2^6\mathrm{mod}127 \\ =64\mathrm{mod}127 \\ =64 \end{gathered}$$

So,$2^{125}\mathrm{mod}127$ is equal to 64.