Question

What is the least significant decimal digit of ${\left({17}^{17}\right)}^{\mathbf{17}}$? (Hint: For distinct primesp,q, and any a is not equal to role="math" localid="1658726105638" $\mathbf{a}\mathbf{\not\equiv }\mathbf{0}\left(\mathrm{mod}\mathrm{pq}\right)$, we proved the formula role="math" localid="1658726171933" ${\mathbf{a}}^{(p-1)}\mathbf{\equiv }\mathbf{1}\mathbf{(}\mathbf{mod}\mathbf{}\mathbf{pq}\mathbf{)}$in Section 1.4.2.)

Short answer

The least significant decimal digit can be categorized as the rightmost digit of the decimal number written in scientific notation having mod of 10 .

Step-by-step solution

Calculate (1717)17 mod 1

We have to find least significant digit of ${\left({17}^{17}\right)}^{17}$.

For that we need to find out modulo of ${\left({17}^{17}\right)}^{17}\mathrm{mod}10$.

Factors of 10 are,1,2,5,10 .

Prime number between the factor of 10 are 2 and 5.10=2,5

Formula we need to use is,

${\mathrm{a}}^{\left(\mathrm{p}-1\right)\left(\mathrm{q}-1\right)}=1\left(\mathrm{mod}\mathrm{pq}\right)$

Calculations

Given that,

a=17,p=2 , and q=5.

By putting these value in the formula, we get

$$\begin{gathered} {17}^{\left(2-1\right)\left(5-1\right)}=1\mathrm{mod}10 \\ {17}^4=\mathrm{mod}10 \end{gathered}$$

${17}^{17}$can also be written as

${17}^{17}={\left(4^2+1\right)}^{17}$

So, ${\left({17}^{17}\right)}^{17}\mathrm{mod}10$can be written as,

${17}^{4\times \mathrm{c}}\left(\mathrm{mod}10\right)\times 17\left(\mathrm{mod}10\right)=7$

Where, C is constant.

The least significant decimal digit of ${\left({17}^{17}\right)}^{17}$ is 7.