special cases of integral solution

Special cases of Integral Solution

case (1): a + b + c = n condition a>b>c≥0

Number of possible intergral solutions
= $\left[ {\frac{{{n^2} + 6}}{{12}}} \right]$ where [x] → Greatest integer function

case (2): a + b + c = n condition a≥b≥c≥0
Number of possible integral solutions
= $\left[ {\frac{{{n^2} + 6}}{{12}}} \right] + \left[ {\frac{n}{2}} \right] + 1$ where [x] → Greatest integer function

case (3): |x| + |y| = n
Number of integral solution = 4n

case (4): |x| + |y| < n
Number of integral solution ⟹ 1 + 4(1 + 2 + 3 + …… + (n-1))

case (5): |x| + |y| ≤ n
Number of integral solution ⟹ 1 + 4(1 + 2 + 3 + 4 + …….. + n)

case (6): |x| + |y| + |z| = n
Number of integral solution ⟹ 4n² + 2

case (7): |x| + |y| + |z| + |w| = n
Number of integral solution ⟹ $\frac{{8n}}{3}.\left( {{n^2} + 2} \right)$

case (8): x * y = n

If n is NOT a perfect square then number of positive integral solutions = $\frac{{\left( {number\,of\,factors\,of\,n} \right)}}{2}$
& number of total integral solutions = 2*(number of positive integral solutions)
= 2* $\frac{{\left( {number\,of\,factors\,of\,n} \right)}}{2}$

If n is a perfect suqare
then number of positive integral solutions= $\frac{{\left( {number\,of\,factors\,of\,n\, + \,1} \right)}}{2}$

total integral soluions = 2* $\frac{{\left( {number\,of\,factors\,of\,n\, + \,1} \right)}}{2}$

case (9): x*y*z = n

Let n = ${a^{{p_1}}}*{b^{{p_2}}}*{c^{{p_3}}}*..............$
then number of ORDERED POSITIVE integral solutions =

& number of ORDERED integral solutions = 4 * number of ordered positive integral solutions

case (10): ax + by = n

⟹ First, reduce the given equation in lowest reducible form.
⟹ After reducing, if coefficients of x and y still have a factor in common, the equation will have no solutions.
⟹ if x & y are co-prime in lowest reducible form, find one solution of x. Then the other solutions can be listed as an arithmetic progression with common difference as the co-efficient of y which can be easily counted.
⟹ if equation is of the form ax+by=n then increase in x as coefficient of y will cause decrease in coefficint of y as a step of coefficient of x.
⟹ if equation is of the form ax-by=n then increase in x as a step of coefficient of y will cause an increase in coefficient of y as a step of coefficient of x.

⟹ Shortcut:→ if for ax+by=n either of a or b can divide n, then number of non negative integral solutions = $\left[ {\frac{n}{{L.C.M.(a,b)}}} \right] + 1$

Let us understand this concept by example:-
Example: Find positive integral solution of 2x+3y = 30
Solution:-

x = 0 ⟶ y = 10 ❌

x = 1 ⟶ y = $\frac{{28}}{3}$ ❌

x = 2 ⟶ y = $\frac{{26}}{3}$ ❌

x = 3 ⟶ y = $\frac{{24}}{3}$ = 8 ✅

x = 6 ⟶ y = 6

x= 9 ⟶ y = 4

x = 12 ⟶ y = 2

x = 15 ⟶ y = 0 ❌ ⟶ not positive

So there are total 4 positive integral solutions.
& there are total 6 non-negative integral solutions of above equation as case( x=15, y=0) & (x=0, y=10) will also be counted.

Example:- Find number of positive & non-negative integral solution of equation 2x + 3y = 57
Solution:-

x = 0 ⟶ y = $\frac{{57}}{3}$ = 19

x = 1 ⟶ y = $\frac{{55}}{3}$ ❌

x = 2 ⟶ y = $\frac{{53}}{3}$ ❌

x = 3 ⟶ y = $\frac{{51}}{3}$ = 17

x = 4 ⟶ y = $\frac{{49}}{3}$ ❌

x = 5 ⟶ y = $\frac{{47}}{3}$ ❌

x = 6 ⟶ y = $\frac{{45}}{3}$ = 15

x = 9 ⟶ y = 13

x = 12 ⟶ y = 11

x = 15 ⟶ y = 9

x = 18 ⟶ y = 7

x = 21 ⟶ y = 5

x = 24 ⟶ y = 3

x = 27 ⟶ y = 1

x = 30 ⟶ y = -1 ❌

Hence total number of non-negative integral solution = 10
& total number of positive integral solution = 9

Shortcut Since 57 is divisible by 3
Hence number of non-negative integral solution
= $\left[ {\frac{{57}}{{LCM(2,3)}}} \right] + 1$

= $\left[ {\frac{{57}}{6}} \right] + 1$

= 9 + 1

= 10 answer

case (11): $\frac{a}{x} \pm \frac{b}{y} = \frac{1}{n}$

$\frac{a}{x} + \frac{b}{y} = \frac{1}{n}$

F = factors of (a*b*n²)
Total integral solution = 2F – 1
Total positive integral solution = F
Total negative integral solution = 0

$\frac{a}{x} - \frac{b}{y} = \frac{1}{n}$

F = number of factors of (a*b*n²)
Total integral solution = 2F – 1
Total positive integral solution = $\frac{{(F - 1)}}{2}$

Total negative integral solution = $\frac{{(F - 1)}}{2}$

Example:- Find total integral & positive solution of equation $\frac{{16}}{x} + \frac{5}{y} = \frac{1}{3}$
Solution:-

F = number of factors of (16×5×3²)
= number of factors of (2⁴×3²×5¹)

F = 5×3×2
= 30
∴ total integral solution = 2F – 1 = 60 – 1 = 59
& total positive integral solution = F = 30 Answer

Example:- Find total integral, positive & negative solution of equation $\frac{4}{x} - \frac{1}{y} = \frac{1}{5}$
Solution:-

F = number of factors of (4 * 1 * 5²)
= number of factors of (2² * 52)
= 3×3 = 9

∴ total integral solution = 2F – 1 = 18 – 1 = 17
total positive integral solution = $\frac{{(F - 1)}}{2}$ = $\frac{{(9 - 1)}}{2}$ = 4

total negative integral solution = $\frac{{(F - 1)}}{2}$ = $\frac{{(9 - 1)}}{2}$ = 4 Answer

special cases of integral solution

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