Section1.2.htm

Section 1.2

2. (a) False. Consider the case where $n=2$ : there is no integer $k$ such that $1=k\cdot 2.$

(b) True. For any positive integer $n$ , $n=1\cdot n.$

(c) True. For any positive integer $n,$ $n^{2}=n\cdot n.$

3. (a)

(b)

4. We check that for each pair $\left( m,n\right)$ in exercise 3,

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13. (a) These are the odd positive integers.

(b) These are the integers that are not multiples of $3.$

(c) For the integer $m$ to be relatively prime to 4 means that $\gcd (m,4)=1$ .

These are precisely the odd positive integers. The reason is this: There are only three possibilities,

namely that $\gcd (m,4)=1,$ $\gcd (m,4)=2,$ or $\gcd (m,4)=4.$ If $m$ is even then one of the last two situations

must happen. If $m$ is odd then the first must happen.

16. (a) and $29$ are relatively prime to $30$ .

(b) and $35$ are relatively prime to $36.$

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