A Nice Diophantine Equation | Integer Solutions

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a+3ab+b=23
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SIMILAR PROBLEM 🤩

PLAYLISTS 🎵 :

[mcwulf]

I would have guessed there was a way to get straight to a+2b without solving for a and b first.

[vladimirkaplun]

Straightforward sol: a=(23-b)/(1+3b)=23-70b(1+3b) - integer. b and 3b+1 are coprime .Hence 1+3b is a divisor of 70...

[InnocentNeuron]

Two ordered pairs will solve it (a, b) == (3, 2) and (2, 3). Accordingly 2 possible solutions are 7 and 8.

[XLatMaths]

I got 7 and 8 from the thumbnail but nice to see the full solution done properly!

[mathswan]

a+2b= 7 or 8 or 23 or 46 or -14 or -25 or -12 or -9

[allanmarder]

Solve for b in terms of a: b=(23-a)/(3a+1). divide top & bottom by a b=[(23/a) -1]/[(3+(1/a)] abs(b) <1 if a<-13 & a>5. Check (23-a)/(3a+1) for -13<a<6.
The following are solutions: (a, b): ( -12, -1) ( -5, -2) ( -1, -12) (0, 23) (2, 3) (3, 2) a+2b: -14, -9, -25, 46, 8, 7

[honestadministrator]

3 a + 9 a b + 3 b = 70
( 1 + 3 a)( 1 + 3 b) = 70
case I : 1 + 3 a = 1, 1 + 3 b = 70
a = 0, b = 23 i.e a + 2 b = 46
case II : 1 + 3 a = 10, 1 + 3 b = 7
a = 3, b = 2 i.e a + 2 b = 7
Extension of this argument gives other two feasible solutions i.e
a = 2, b = 3 and a = 23, b = 0
Hereby a + 2 b = 7, 46

[walterwen]

A Nice Diophantine Equation: a + 3ab + b = 23, a + 2b = ?; Integer solutions
3(a + 3ab + b) = 3(23), 3a(1 + 3b) + 1 + 3b = 69 + 1, (3a + 1)(3b + 1) = 70
Let: m = 3a + 1, n = 3b + 1
(3a + 1)(3b + 1) = 70 = (m)(n) = (± 1)(± 70) = (± 2)(± 35) = (± 5)(± 14) = (± 7)(± 10)
3a = m – 1, 3b = n – 1
1.0 m = 1, 3a = 0, a = 0; n = 70, 3b = 69, b = 23
m = 70, 3a = 69, a = 23; n = 1, 3b = 0, b = 0
a + 2b = 0 + 2(23) = 46 or a + 2b = 23 + 2(0) = 23
1.1 m = – 1, 3a = – 2; Rejected, not integer
m = – 70, 3a = – 71; Rejected, not integer
2.0 m = 2, 3a = 1; Rejected, not integer
m = 35, 3a = 34; Rejected, not integer
2.1 m = – 2, 3a = – 3, a = – 1; n = – 35, 3b = – 36, b = – 12
m = – 35, 3a = – 36, a = – 12; n = – 2, 3b = – 3, b = – 1
a + 2b = – 1 + 2(– 12) = – 25 or a + 2b = – 12 + 2(– 1) = – 14
3.0 m = 5, 3a = 4; Rejected, not integer
m = 14, 3a = 13; Rejected, not integer
3.1 m = – 5, 3a = – 6, a = – 2; n = – 14, 3b = – 15, b = – 5
m = – 14, 3a = – 15, a = – 5; n = – 5, 3b = – 6, b = – 2
a + 2b = – 2 + 2(– 5) = – 12 or a + 2b = – 5 + 2(– 2) = – 9
4.0 m = 7, 3a = 6, a = 2; n = 10, 3b = 9, b = 3
m = 10, 3a = 9, a = 3; n = 7, 3b = 6, b = 2
a + 2b = 2 + 2(3) = 8 or a + 2b = 3 + 2(2) = 7
4.1 m = – 7, 3a = – 8; Rejected, not integer
m = – 10, 3a = – 11; Rejected, not integer
Final answer:
a + 2b = – 25, – 14, – 12, – 9, 7, 8, 23, 46