Algebra Know-It-ALL

594 Worked-Out Solutions to Exercises: Chapters 1 to 9

6. When you break the number 841 into a product of primes using the same process as
   you did to solve Prob. 5, you’ll see that

841 = 29 × 29

It is a perfect square of a prime.

7. When you factor 2,197 into a product of primes (again using the process you did to
   solve Probs. 5 and 6), you’ll see that

2,197= 13 × 13 × 13

It has three prime factors, all equal to 13. When a natural number is multiplied by itself

and then the result is multiplied by the original number again, the product is a perfect

cube. The number 2,197 is a perfect cube of a prime.

8. Any composite number can be factored into a product of primes. In the traditional
   sense, all the primes are positive because they are all natural numbers larger than 1.
   Remember from basic arithmetic that whenever you multiply a positive number times
   another positive number, the result is always positive. That means no negative number
   can be composite if we stick to the traditional definition of a prime number.


9. Suppose 0 were defined as prime. Then 0 would also be composite, because you can
   multiply 0 by any prime you want, and you always get 0. An even more serious problem
   occurs if we let 1 be called prime. If that were true, then 1 times any other prime
   would be equal to that same prime, making every prime number composite! That’s why
   mathematicians generally refuse to call 0 or 1 prime numbers. But when we don’t allow
   them to be prime, they can’t be composite either, because they can’t be broken down
   into factors from the traditional set of primes {2, 3, 5, 7, 11, 13, 17, 19, ...}.


10. In Fig. 3-5, you start with 0 and proceed through the positive and negative integers
    alternately. You can create an “implied one-ended list” of the entire set Z of integers
    starting with 0, one after the other, this way:

Z= {0, 1, −1, 2, −2, 3, −3, ...}

If you pick any integer, no matter how large positively or negatively, this “implied one-ended

list” will eventually get to it. You can pair the set of all natural numbers one-to-one with the

set of all integers like this, with natural numbers on the left and integers on the right:

0 ↔ 0

1 ↔ 1

2 ↔− 1

3 ↔ 2

4 ↔− 2

5 ↔ 3

6 ↔− 3

↓

and so on, forever