General Solution of Diophantine Equation Example || Modular Arithmetic || Dr. Sujata T

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[samueldeandrade]

This is NOT good. Look:

70x+112y = 168

To make 8 appear as unit digit we have to multiply 112 by a number with 4 or 9 as unit digit. With 4 we have

112×4 = 448

Subtrating it from 168,

168-448 = -280

This has to be

70x = -280 → x = -4

So, (-4, 4) is a particular solution. Now, to find the gcd, let's find the prime factors

70 = 2×35 = 2×5×7
112 = 2×56 = 2×8×7 = 2⁴×7

With that we have

gcd(70, 112) = 2×7

The general solutions is simply

x = -4+8k
y = 4-5k

Why complicate things???

[shebint.p]

Hello ma'am how get 14/168 equal 12...?