Property 2
Prime Numbers
Positive numbers greater than 1 which are only divisible by 1 and itself are called prime numbers.
\(2,3,5,7,11,13,17,\ldots\)
Relatively Prime Numbers
Numbers which do not have a common positive factor other than 1 are called relatively prime numbers.
(For example 8 and 15 are relatively prime numbers)
- When determining the minimum or maximum value of the expression, the kinds of the number must be taken into consideration. Also, in equations with integers the coefficients of the expression are made relatively prime.
1.
\(a,b\in\mathbb{Z}^{+}\)
\(a\ne b\)
\(\Rightarrow\min(2a+3b)=?\)
2.
\(a,b\in\mathbb{N}\)
\(a\ne b\)
\(\Rightarrow\min(3a+4b)=?\)
3.
\(a,b,c\in\mathbb{Z}^{+}\)
\(\Rightarrow\min(2a+3b+4c)=?\)
4.
\(a,b,c\in\mathbb{Z}^{-}\)
\(a\ne b\ne c\)
\(\Rightarrow\max(7a+b+3c)=?\)
5.
\(a,b,c\in\mathbb{Z}^{-}\)
\(\Rightarrow\max(3a+2b+6c)=?\)
6.
\(a,b,c\in\mathbb{Z}^{+}\)
\(\Rightarrow\min(3a+2b+6c)=?\)
7.
\(a,b,c\in\mathbb{Z}^{+}\)
\(a\ne b\ne c\)
\(\Rightarrow\min(a+b+3c)=?\)
8.
\(a,b\in\mathbb{Z}^{+}\)
\(4\cdot a=5\cdot b\)
\(\Rightarrow \min(a+b)=?\)
9.
\(a,b\in\mathbb{Z}^{+}\)
\(15\cdot a=12\cdot b\)
\(\Rightarrow \min(a+b)=?\)
10.
\(a,b\in\mathbb{Z}^{-}\)
\(20\cdot a=15\cdot b\)
\(\Rightarrow \max(a+b)=?\)
11.
\(a,b,c\in\mathbb{Z}^{+}\)
\(2\cdot a=3\cdot b\)
\(5\cdot b=4\cdot c\)
\(\Rightarrow\min(a+b+c)=?\)
12.
\(a,b,c\in\mathbb{Z}^{+}\)
\(a\cdot b=18\)
\(b\cdot c=21\)
\(\Rightarrow\min(a+b+c)=?\)
13.
\(a,b,c\in\mathbb{Z}^{+}\)
\(a\cdot b=18\)
\(b\cdot c=21\)
\(\Rightarrow\max(a+b+c)=?\)