Question 3-7
In the set N, what is a perfect square? Can any perfect square be prime?
Answer 3-7
A perfect square is the result of multiplying a natural number by itself. The first few perfect
squares are 0, 1, 4, 9, 16, 25, 36, 49, and 64. No perfect square can be prime. By definition,
0 and 1 are not prime. Any perfect square larger than 1 can be broken down into a product of
two or more primes, so it’s composite.
Question 3-8
How can we write down the set Z of integers as an “implied, two-ended list”? As an “implied,
one-ended list”?
Answer 3-8
Here’s an “implied, two-ended list” that can give any reader the basic idea concerning the
elements of Z:
Z= {..., −4,−3,−2,−1, 0, 1, 2, 3, 4, ...}To create the “implied, one-ended list,” we start with 0 and then go through the positive and
negative integers alternately, like this:
Z= {0, 1, −1, 2, −2, 3, −3, 4, −4, ...}Question 3-9
How can we quickly and easily tell if a large natural number is divisible by 2, 3, 5, 9, or 10
without a remainder?
Answer 3-9
A natural number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. A natural number is divis-
ible by 3 if the sum of its digits is a natural-number multiple of 3, by 5 if its numeral ends in
0 or 5, by 9 if the sum of its numeric digits is a natural-number multiple of 9, and by 10 if its
numeral ends in 0.
Question 3-10
How can we find the prime factors of a large natural number n?
Answer 3-10
We start by finding the square root of n. We ignore the digits after the decimal point, so we
have a whole number. We add 1 to that whole number and call the result s. Then we
divide the original number n by all the primes less than or equal to s, one by one, starting
with the largest prime and working our way down. If we ever get a whole-number quotient,
then we know that the divisor and the quotient are both factors of n. Sometimes the quotient
is prime, and sometimes it is not. If it isn’t prime, then it can be factored down further. We
keep dividing n by smaller and smaller primes until we get down to 2. Once we’ve found all
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