Answer:
True; this [tex]x[/tex] would indeed be a factor of [tex]z[/tex].
Step-by-step explanation:
"[tex]x[/tex] is a factor of [tex]y[/tex]" means that there is a whole number [tex]p[/tex] such that [tex]x\, p = y[/tex].
Likewise, [tex]y[/tex] is a factor of [tex]z[/tex] if and only if there is a whole number [tex]q[/tex] such that [tex]y\, q = z[/tex].
This [tex]x[/tex] would be a factor of [tex]z[/tex] if there is some whole number [tex]r[/tex] such that [tex]x\, r = z[/tex].
The first equality [tex]x\, p = y[/tex] implies that [tex]y[/tex] may be substituted with [tex]x\, p[/tex].
Substitute the [tex]y[/tex] in the second equality [tex]y\, q = z[/tex] with [tex](x\, p)[/tex] to obtain:
[tex](x\, p)\, q = z[/tex].
Rearrange to obtain:
[tex]x\, (p\, q) = z[/tex].
The product of two whole numbers is still a whole number. Since [tex]p[/tex] and [tex]q[/tex] are both whole numbers, [tex]p\, q[/tex] would also be a whole number.
Thus, if [tex]r = p\, q[/tex], then [tex]x\, r = x\, (p\, q) = z[/tex], while [tex]r[/tex] would be a whole number.
Therefore, this [tex]x[/tex] would indeed be a factor of [tex]z[/tex].
