Let's go through the expressions step-by-step, as well as verifying that each set of numbers can be expressed as a multiple of their sum with no common factor.
### Expressions as Multiples of Sums
1. Expression: [tex]\(27 \times (2 + 3 + 1)\)[/tex]
- Sum of numbers: [tex]\(2 + 3 + 1 = 6\)[/tex]
- Representation: [tex]\(27 \times 6\)[/tex]
2. Expression: [tex]\(9 \times (2 + 4 + 3)\)[/tex]
- Sum of numbers: [tex]\(2 + 4 + 3 = 9\)[/tex]
- Representation: [tex]\(9 \times 9\)[/tex]
3. Expression: [tex]\(22 \times (4 + 3 + 1)\)[/tex]
- Sum of numbers: [tex]\(4 + 3 + 1 = 8\)[/tex]
- Representation: [tex]\(22 \times 8\)[/tex]
4. Expression: [tex]\(12 \times (3 + 8 + 1)\)[/tex]
- Sum of numbers: [tex]\(3 + 8 + 1 = 12\)[/tex]
- Representation: [tex]\(12 \times 12\)[/tex]
5. Expression: [tex]\(14 \times (9 + 2 + 1)\)[/tex]
- Sum of numbers: [tex]\(9 + 2 + 1 = 12\)[/tex]
- Representation: [tex]\(14 \times 12\)[/tex]
6. Expression: [tex]\(4 \times (4 + 7 + 12)\)[/tex]
- Sum of numbers: [tex]\(4 + 7 + 12 = 23\)[/tex]
- Representation: [tex]\(4 \times 23\)[/tex]
### Predefined Numbers
Now let’s list the predefined numbers once more and verify them:
1. Predefined Sum: [tex]\(16 + 28 + 48 = 92\)[/tex]
2. Predefined Sum: [tex]\(126 + 28 + 14 = 168\)[/tex]
3. Predefined Sum: [tex]\(36 + 96 + 12 = 144\)[/tex]
4. Predefined Sum: [tex]\(88 + 66 + 22 = 176\)[/tex]
5. Predefined Sum: [tex]\(54 + 81 + 27 = 162\)[/tex]
6. Predefined Sum: [tex]\(18 + 36 + 27 = 81\)[/tex]
### Summary of Results
Here are the results for the expressions as multiples of sums of numbers, along with the predefined sums:
- Expressions as multiples of sums:
[tex]\[
\begin{align*}
(27, 6) \\
(9, 9) \\
(22, 8) \\
(12, 12) \\
(14, 12) \\
(4, 23)
\end{align*}
\][/tex]
- Predefined sums:
[tex]\[
\begin{align*}
16 + 28 + 48 = 92 \\
126 + 28 + 14 = 168 \\
36 + 96 + 12 = 144 \\
88 + 66 + 22 = 176 \\
54 + 81 + 27 = 162 \\
18 + 36 + 27 = 81
\end{align*}
\][/tex]
These sums confirm the totals given in the question. Therefore, the representation is verified to be correct.
