Work Sheet #1 Q1: Use the definitions of even and odd to justify your answers to

Question

Work Sheet #1

Q1: Use the definitions of even and odd to justify your answers to the following questions:

Definition 1:

An integer n is even if and only if n equals twice some integer. An integer n is odd if and only if n equals twice some integer plus 1.

Symbolically, if n is an integer, then

n is even    an integer k such that n = 2k

n is odd    an integer k such that n = 2k+1

1.) Is 0 even?

2.) Is 301 odd?

3.) If a and b are integers, is 6a2b even?

4.) If a and b are integers, is 10a + 8b + 1 odd?

5.) Is every integer either even or odd?

Q2: Use the definitions of prime numbers to justify your answers to the following questions

Definition 2:

An integer n is a prime if and only if, n > 1 and for all positive integers r and s, if n = rs, then either r or s equals n. An integer n is composite if and only if n > 1 and n = rs for some integers r and s with 1<r<n and 1<s<n.

In symbols:

n is prime   for all positive integers r and s, if n=rs then either r =1 and s=n or r=n and s=1.

n is composite    positive integers r and s such that n=rs and 1<r<n and 1<s<n.

a. Is 1 prime?

b. Is every integer greater than 1 either prime or composite?

c. Write the first six prime numbers.

d. Write the first six composite numbers.

Explanation / Answer

Q1.) As per Chegg policy, I am answering only first question

1.) As per definition, 0 is even as 0 = 2*0

2.) -301 is odd as -301 = 2*(-151) + 1

3.) 6a2b can be even or odd, depending on the value of b. If b is even then 6a2b is also even otherwise odd.

4.) Yes. 10a+8b+1 is odd. As multiplication with 10 and 8 makes the number 1 and the addition of 1 makes it odd.

5.) Yes, with being 0 as even integer. Every number can be classified as either even or odd

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