Unformatted text preview: 10 12. 13. 14. 15. Chapter 1 Divisibility and Factorization (e) Prove that the sum of an even integer and an odd integer is odd and
that their product is even. Prove that the square of any odd integer is expressible in the form 8n + 1 with n e Z. Prove that the fourth power of any odd integer is expressible in the form 16n + 1 with rt EZ. (a) Let x be a positive real number and let d be a positive integer. Prove
that the number of positive integers less than or equal to x that are
divisible by d is [i]. (b) Find the number of positive integers not exceeding 500 that are
divisible by 3. (c) Find the number of positive integers between 200 and 500 that are
divisible by 3. [The following exercise presents two alternate versions of the division algorithm (Theorem 1.4). Both versions allow negative divisors; as such, they are more general than Theorem 1.4.] (a) Let a and b be nonzero integers. Prove that there exist unique q, r e Z
such that a=bq+r, 05r<|bl (b) Find the unique q and r guaranteed by the division algorithm of part
(a) above with a = 47 and b : —6.
(c) Let a and b be nonzero integers. Prove that there exist unique q, r e Z such that
:b + ——< £7
a q r, 2 r 2 This algorithm is called the absolute least remainder algorithm.
(d) Find the unique q and r guaranteed by the division algorithm of part
(0) above with a = 47 and b = —6. __-__ Prime Numbers Every integer greater than one has at least two positive divisors, namely, 1 and
the integer itself. Those positive integers having no other positive divisors (and
so exactly two positive divisors) are of crucial importance in number theory
and are introduced now. Deﬁnition 5: Let p e Z with p > 1. Then p is said to be prime if the only positive divisors ofp are 1 and p. If rt 6 Z, rt > 1, and rt is not prime, then H is
said to be composite. Note that the positive integer 1 is neither prime nor composite by deﬁnition.
The reason for disallowing 1 as a prime number is investigated in Exercise 66. ...
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