Question

Product of ab and c is de. The sum of de and fg is hi. Find a,b,c,d,e,f,g,h,i and every letters are from1-9 by using one number only once

Answer 1

To solve the problem where the letters represent unique digits from 1 to 9, we need to find values for \(a, b, c, d, e, f, g, h, i\) under the constraints:

1. **Product**: \( ab \times c = de \)
2. **Sum**: \( de + fg = hi \)

Let's break down the problem systematically. Here, \(ab\) represents a two-digit number where \(a\) is the tens place and \(b\) is the units place, and \(de\) represents another two-digit number similarly.

### Steps to solve:

1. **Enumerate all possible values of ab and c**:
- The maximum value for \(ab\) (from 10 to 99) and \(c\) (from 1 to 9).
- This means the maximum product of \(ab \times c\) can be \(99 \times 9 = 891\), which is greater than the largest allowed two-digit number 99.
- Thus, combinations should lead to \(de\) being another valid two-digit number.

2. **Calculate de**:
- Since \(de\) results from \(ab \times c\), it has to be less than or equal to a two-digit number (99).

3. **Calculate fg and hi**:
- \(fg\) can take values ranging from 10 to 99.
- From the sum equation, \(de + fg = hi\) must also yield a two-digit number.

### Implementation:
Here’s a programmatic way you could approach the solution:

1. Iterate through values for \(a, b, c, d, e, f, g\) ensuring they each meet the criteria of using distinct digits from 1 to 9.
2. For every valid combination of \(a\), \(b\), and \(c\) that satisfies the product \(ab \times c = de\), check combinations for \(fg\) that yield \(hi = de + fg\).
3. Ensure all combinations are different digits.

### Continuing the search manually:
Considering smaller combinations to check feasibly:

1. Assume \(ab = 12\) and \(c = 3 \Rightarrow de = 12 \times 3 = 36\)

Now check \(fg\):
- Let’s assume \(fg = 24\): \(36 + 24 = 60\) so \(hi = 60\).

Then we determine if they fit:

- \(a = 1\), \(b = 2\), \(c=3\), \(d=3\), \(e=6\) (which is a repeat)
Thereby we'd need to explore different possible values.

After trying multiple combinations, we arrive at a valid solution:

- \(a = 1, b = 5, c = 7\)
- \(de = 1 \times 5 \times 7 = 35\)

We can try more:

After several iterations, we find:

- \(a = 3, b = 4, c = 2 \Rightarrow 34 \times 2 = 68\)

Assuming checking distinct pairs of \(fg\):

If \(fg = 19\)

Then,

- The sum \(68 + 19 = 87\)

### Final Assignment:
Thus:

- \(a=3\)
- \(b=4\)
- \(c=2\)
- \(d=6\)
- \(e=8\)
- \(f=1\)
- \(g=9\)
- \(h=8\)
- \(i=7\) would make valid assignments and constructs.

Continue iterating this way until every number is distinct and fits the constraints.

This algorithm may naturally converge or singular digits used throughout. The critical part is ensuring no repeats, valid constructs and checking sum correctness.

Note: This play around a methodical search and combinatorial evaluations leads toward systematic searching possible variants.