Chapter 1: Q26E (page 49)

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

Short Answer

Expert verified

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 ${(17^{17})}^{17}$ .

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

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,

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

Calculations

Given that,

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

By putting these value in the formula, we get

$17^{(2 - 1) (5 - 1)} = 1 \mod 10 17^{4} = \mod 10$

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

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

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

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

Where, C is constant.

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

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

Short Answer

Step by step solution

Calculate (1717)17 mod 1

Calculations

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What is the least significant decimal digit of (1717)17? (Hint: For distinct primesp,q, and any a is not equal to role="math" localid="1658726105638" a≢0(modpq), we proved the formula role="math" localid="1658726171933" a(p-1)≡1(modpq)in Section 1.4.2.)

Short Answer

Step by step solution

Calculate (1717)17 mod 1

Calculations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Computer Science Textbooks

Data Structures

Databases

Issues in Computer Science

Theory of Computation

Data Representation in Computer Science

Computer Systems

Study anywhere. Anytime. Across all devices.

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