Given: [tex]a, b \in \mathbb{Z}[/tex].
If in each case [tex]S \in \mathbb{Z}[/tex], simplify the expression for [tex]S[/tex] and indicate the result used at each step. Then, calculate [tex]S[/tex] using the given values of [tex]a[/tex] and [tex]b[/tex].
a) [tex]S = (30a)(-40b)(10a)(-20b)[/tex], where [tex]a = 1[/tex] and [tex]b = -1[/tex].
1. Simplify the expression:
[tex]S = (30a)(-40b)(10a)(-20b)[/tex]
2. Substitute the given values:
[tex]S = (30 \cdot 1)(-40 \cdot -1)(10 \cdot 1)(-20 \cdot -1)[/tex]
3. Calculate the product:
[tex]S = (30)(40)(10)(20)[/tex]
4. Final result:
[tex]S = 30 \cdot 40 \cdot 10 \cdot 20 = 240,000[/tex]