Diophantine Equation Solutions: 100D+50H+25Q+10d+5n=500 (Positive Integers)

Finds all positive integer solutions for the linear Diophantine equation 100D+50H+25Q+10d+5n=500 by systematically reducing the variables and iterating through possible combinations.

This is a linear Diophantine equation with five variables, subject to the constraint that all variables must be positive integers.

The equation is:

where

Since all coefficients are divisible by 5, we can divide the entire equation by 5:

We need to find all positive integer solutions

We can analyze the possible values for the variables with the largest coefficients first, which are

, and

Step 1: Analyze the bounds for D, H, and Q.

Since

:
The minimum value of the remaining terms is:

The equation can be rewritten as:

Since

, we have:

Also, since

A. Determine possible values for D

Since

.
If

, impossible since

.
So, for

.
If

, impossible.
So, for

.
If

, impossible.
So, for

.
If

, impossible.
So, for

and

Since

, the LHS is strictly positive. No solutions for

We must systematically check the combinations

and then solve for

Let

. We need

, with

.
Since

, we require

, so

, or

.
Since

, we need

, which means

, or

. This condition is already incorporated in our bounding analysis (

must be at least 3).
For a fixed

, the number of solutions for

Case D=1 ()

. We need

1.1. ()

.
For a fixed

. Number of solutions for

Q	R		Solutions (d, n)
1	65	32	(1, 63), (2, 61), …, (32, 1)
2	60	29	(1, 58), (2, 56), …, (29, 2)
3	55	27	(1, 53), (2, 51), …, (27, 1)
4	50	24	(1, 48), …, (24, 2)
5	45	22	(1, 43), …, (22, 1)
6	40	19	(1, 38), …, (19, 2)
7	35	17	(1, 33), …, (17, 1)
8	30	14	(1, 28), …, (14, 2)
9	25	12	(1, 23), …, (12, 1)
10	20	9	(1, 18), …, (9, 2)
11	15	7	(1, 13), …, (7, 1)
12	10	4	(1, 8), …, (4, 2)
13	5	2	(1, 3), (2, 1)

Total solutions for

1.2. ()

Q	R
1	55	27
2	50	24
3	45	22
4	40	19
5	35	17
6	30	14
7	25	12
8	20	9
9	15	7
10	10	4
11	5	2

Total solutions for

1.3. ()

Q	R
1	45	22
2	40	19
3	35	17
4	30	14
5	25	12
6	20	9
7	15	7
8	10	4
9	5	2

Total solutions for

1.4. ()

Q	R
1	35	17
2	30	14
3	25	12
4	20	9
5	15	7
6	10	4
7	5	2

Total solutions for

1.5. ()

Q	R
1	25	12
2	20	9
3	15	7
4	10	4
5	5	2

Total solutions for

1.6. ()

Q	R
1	15	7
2	10	4
3	5	2

Total solutions for

1.7. ()

Q	R
1	5	2

Total solutions for

: 2. (Solutions:

)

Total solutions for D=1:

Case D=2 ()

. We need

2.1. ()

This is identical to Case 1.3. Total solutions: 106.

2.2. ()

This is identical to Case 1.4. Total solutions: 65.

2.3. ()

This is identical to Case 1.5. Total solutions: 34.

2.4. ()

This is identical to Case 1.6. Total solutions: 13.

2.5. ()

This is identical to Case 1.7. Total solutions: 2. (Q must be 1).

Total solutions for D=2:

Case D=3 ()

. We need

3.1. ()

This is identical to Case 1.5. Total solutions: 34.

3.2. ()

This is identical to Case 1.6. Total solutions: 13.

3.3. ()

This is identical to Case 1.7. Total solutions: 2. (Q must be 1).

Total solutions for D=3:

Case D=4 ()

We established that

and

Since

:
If

. Solution:

.
If

. Solution:

.
If

, impossible.
Total solutions for

: 2. (The solutions are

and

Conclusion

Total number of possible solutions is the sum of solutions from all cases:
Total

Total

The problem asks to “Find all possible solutions.” Since the number of solutions is large (863), listing them all is impractical. The standard interpretation for such problems when the count is large is to provide the general method or the total count, or to list representative examples if constraints allow for a manageable set. Given the context, listing the method and the total count is the most appropriate formal answer.

The set of all possible solutions

consists of 863 unique tuples of positive integers satisfying the equation.

Example solutions:

. . Solutions: and .
Check : . (Correct)
. . One solution is . .
Check: . (Correct)

The total number of solutions is 863.

Diophantine Equation Solutions: 100D+50H+25Q+10d+5n=500 (Positive Integers)

Question

Answer