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Chapter 4: Problem 1

Determine whether each of these integers is prime. $$\begin{array}{ll}{\text { a) } 21} & {\text { b) } 29} \\ {\text { c) } 71} & {\text { d) } 97} \\ {\text { e) } 111} & {\text { f) } 143}\end{array}$$

Short Answer

Expert verified
21: not prime, 29: prime, 71: prime, 97: prime, 111: not prime, 143: not prime

Step by step solution

01

- Understanding Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
02

- Check number 21

Check if 21 is divisible by any integers other than 1 and 21. Since 21 is divisible by 3 (21 梅 3 = 7), 21 is not a prime number.
03

- Check number 29

Check if 29 is divisible by any integers other than 1 and 29. Since 29 is not divisible by any other numbers except 1 and 29, 29 is a prime number.
04

- Check number 71

Check if 71 is divisible by any integers other than 1 and 71. Since 71 is not divisible by any other numbers except 1 and 71, 71 is a prime number.
05

- Check number 97

Check if 97 is divisible by any integers other than 1 and 97. Since 97 is not divisible by any other numbers except 1 and 97, 97 is a prime number.
06

- Check number 111

Check if 111 is divisible by any integers other than 1 and 111. Since 111 is divisible by 3 (111 梅 3 鈮� 37), 111 is not a prime number.
07

- Check number 143

Check if 143 is divisible by any integers other than 1 and 143. Since 143 is divisible by 11 (143 梅 11 鈮� 13), 143 is not a prime number.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Numbers
Natural numbers are the set of positive integers starting from 1. They are the most basic numbers used for counting and ordering. Examples include 1, 2, 3, 4, and so on.

Every natural number greater than 1 is either a prime number or can be explained as a product of prime numbers. Ensuring you understand natural numbers is crucial as it forms the basis for understanding more complex mathematical concepts like prime numbers.
Divisors
A divisor of a number is any number that divides another number completely without leaving a remainder. For example, the numbers 1, 2, 3, and 6 are divisors of 6 because 6 divided by each of these numbers results in an integer.

Recognizing divisors is essential in determining whether a number is prime or not.
  • If a number has exactly two distinct positive divisors, 1 and itself, it's a prime number.
  • If a number has more than two divisors, it is not a prime number.
When checking for divisors, start with the smallest prime numbers and work your way up.
Prime Number Checking
Prime number checking involves verifying if a number has any divisors other than 1 and itself. If such divisors are found, the number is not prime.

Here are the steps to check if a number is prime:
  • **Step 1:** Check divisibility by the smallest prime numbers (2, 3, 5, etc.). For example, to check if 29 is prime, see if it can be divided evenly by 2, 3, or 5.
  • **Step 2:** Continue this process with each subsequent prime number up to the square root of the number you're checking. If no divisors are found, the number is prime.
For example, to check if 97 is prime:
  • 97 is not divisible by 2 (since it's odd).
  • 97 divided by 3 doesn鈥檛 yield an integer.
  • 97 divided by 5 doesn鈥檛 yield an integer.
  • Consistent checks with other primes up to approximately 9.8 (square root of 97) show that 97 has no divisors other than 1 and itself.
Therefore, 97 is a prime number. This method ensures you correctly determine whether a number is prime by systematically excluding potential divisors.

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Most popular questions from this chapter

What are the greatest common divisors of these pairs of integers? a) \(3^{7} \cdot 5^{3} \cdot 7^{3}, 2^{11} \cdot 3^{5} \cdot 5^{9}\) b) \(11 \cdot 13 \cdot 17,2^{9} \cdot 3^{7} \cdot 5^{5} \cdot 7^{3}\) c) \(23^{31}, 23^{17}\) d) \(41 \cdot 43 \cdot 53,41 \cdot 43 \cdot 53\) e) \(3^{13} \cdot 5^{17}, 2^{12} \cdot 7^{21}\) f) \(1111,0\)

A parking lot has 31 visitor spaces, numbered from 0 to \(30 .\) Visitors are assigned parking spaces using the hashing function \(h(k)=k\) mod \(31,\) where \(k\) is the number formed from the first three digits on a visitor's license plate. a) Which spaces are assigned by the hashing function to cars that have these first three digits on their license plates: \(317,918,007,100,111,310 ?\) b) Describe a procedure visitors should follow to find a free parking space, when the space they are assigned is occupied. Another way to resolve collisions in hashing is to use double hashing. We use an initial hashing function \(h(k)=k \bmod p,\) where \(p\) is prime. We also use a second hashing function \(g(k)=(k+1) \bmod (p-2) .\) When a collision occurs, we use a probing sequence \(h(k, i)=(h(k)+i \cdot g(k)) \bmod p .\)

Which memory locations are assigned by the hashing function \(h(k)=k \bmod 101\) to the records of insurance company customers with these Social Security numbers? $$ \begin{array}{ll}{\text { a) } 104578690} & {\text { b) } 43222187} \\ {\text { c) } 372201919} & {\text { d) } 501338753}\end{array} $$

What is the decryption function for an affine cipher if the encryption function is \(c=(15 p+13) \bmod 26 ?\)

Devise an algorithm that, given the binary expansions of the integers \(a\) and \(b\) , determines whether \(a>b, a=b,\) of \(a
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