6. Find two monomials with positive integer coefficients whose product is the given monomial.

(i) abc
- Examples: [tex]$a$[/tex] and [tex]$bc$[/tex], [tex]$b$[/tex] and [tex]$ac$[/tex], [tex]$c$[/tex] and [tex]$ab$[/tex], [tex]$1$[/tex] and [tex]$abc$[/tex]

(ii) xyz

(iii) [tex]$a^2 bc$[/tex]

(iv) [tex]$7abc$[/tex]

Certainly! To find two monomials whose product is a given monomial, we need to consider the different ways we can split the given monomial into two factors with positive integer coefficients. Let's go through each case step-by-step:

### (i) [tex]$ abc $[/tex]
To factorize [tex]$ abc $[/tex], we look for pairs of monomials whose product is [tex]$ abc $[/tex]. The possible pairs are:

1. [tex]$ a \times bc $[/tex]:
- First monomial: [tex]$ a $[/tex]
- Second monomial: [tex]$ bc $[/tex]
2. [tex]$ b \times ac $[/tex]:
- First monomial: [tex]$ b $[/tex]
- Second monomial: [tex]$ ac $[/tex]
3. [tex]$ c \times ab $[/tex]:
- First monomial: [tex]$ c $[/tex]
- Second monomial: [tex]$ ab $[/tex]
4. [tex]$ 1 \times abc $[/tex]:
- First monomial: [tex]$ 1 $[/tex]
- Second monomial: [tex]$ abc $[/tex]

So, the factor pairs for [tex]$ abc $[/tex] are:
[tex]\[ (a, bc), (b, ac), (c, ab), (1, abc) \][/tex]

### (ii) [tex]$ xyz $[/tex]
To factorize [tex]$ xyz $[/tex], we look for pairs of monomials whose product is [tex]$ xyz $[/tex]. The possible pairs are:

1. [tex]$ x \times yz $[/tex]:
- First monomial: [tex]$ x $[/tex]
- Second monomial: [tex]$ yz $[/tex]
2. [tex]$ y \times xz $[/tex]:
- First monomial: [tex]$ y $[/tex]
- Second monomial: [tex]$ xz $[/tex]
3. [tex]$ z \times xy $[/tex]:
- First monomial: [tex]$ z $[/tex]
- Second monomial: [tex]$ xy $[/tex]
4. [tex]$ 1 \times xyz $[/tex]:
- First monomial: [tex]$ 1 $[/tex]
- Second monomial: [tex]$ xyz $[/tex]

So, the factor pairs for [tex]$ xyz $[/tex] are:
[tex]\[ (x, yz), (y, xz), (z, xy), (1, xyz) \][/tex]

### (iii) [tex]$ a^2 b c $[/tex]
To factorize [tex]$ a^2 b c $[/tex], we look for pairs of monomials whose product is [tex]$ a^2 b c $[/tex]. The possible pairs are:

1. [tex]$ a \times a bc $[/tex]:
- First monomial: [tex]$ a $[/tex]
- Second monomial: [tex]$ a bc $[/tex]
2. [tex]$ a^2 \times bc $[/tex]:
- First monomial: [tex]$ a^2 $[/tex]
- Second monomial: [tex]$ bc $[/tex]
3. [tex]$ b \times a^2 c $[/tex]:
- First monomial: [tex]$ b $[/tex]
- Second monomial: [tex]$ a^2 c $[/tex]
4. [tex]$ c \times a^2 b $[/tex]:
- First monomial: [tex]$ c $[/tex]
- Second monomial: [tex]$ a^2 b $[/tex]
5. [tex]$ 1 \times a^2 b c $[/tex]:
- First monomial: [tex]$ 1 $[/tex]
- Second monomial: [tex]$ a^2 b c $[/tex]

So, the factor pairs for [tex]$ a^2 b c $[/tex] are:
[tex]\[ (a, a bc), (a^2, bc), (b, a^2 c), (c, a^2 b), (1, a^2 b c) \][/tex]

### (iv) [tex]$ 7 a b c $[/tex]
To factorize [tex]$ 7 a b c $[/tex], we look for pairs of monomials whose product is [tex]$ 7 a b c $[/tex]. The possible pairs are:

1. [tex]$ 7 \times a bc $[/tex]:
- First monomial: [tex]$ 7 $[/tex]
- Second monomial: [tex]$ a bc $[/tex]
2. [tex]$ 7a \times bc $[/tex]:
- First monomial: [tex]$ 7a $[/tex]
- Second monomial: [tex]$ bc $[/tex]
3. [tex]$ 7b \times ac $[/tex]:
- First monomial: [tex]$ 7b $[/tex]
- Second monomial: [tex]$ ac $[/tex]
4. [tex]$ 7c \times ab $[/tex]:
- First monomial: [tex]$ 7c $[/tex]
- Second monomial: [tex]$ ab $[/tex]
5. [tex]$ 1 \times 7 a b c $[/tex]:
- First monomial: [tex]$ 1 $[/tex]
- Second monomial: [tex]$ 7 a b c $[/tex]

So, the factor pairs for [tex]$ 7 a b c $[/tex] are:
[tex]\[ (7, a bc), (7a, bc), (7b, ac), (7c, ab), (1, 7 a b c) \][/tex]