The Day I Saw Hidden Vectors Behind 72 and 42

(A little spark for fellow lifelong learners)

Not long ago, while revisiting a math textbook, I followed the classic Euclidean Algorithm to solve a linear Diophantine equation:

72x + 42y = 6

I knew the steps:

72 = 42 + 30

42 = 1×30 + 12

30 = 2×12 + 6

Then I reversed the process — and that’s where the beauty hit me.

Instead of just crunching numbers, I started tracking how each number was built from 72 and 42, step by step:

30 = (1, -1)

12 = (-1, 2)

6 = (3, -5)

These weren’t just coefficients anymore. They felt like 2-D vectors — and even more than that: I was witnessing a transformation of one basis into another. The vectors were shrinking, simplifying — a natural change of perspective. I was no longer carrying 72 and 42 everywhere. I was working with something leaner, smarter.

It was the first time I felt this method as something almost visual — not mechanical.

And that moment made my day.

I now believe moments like this are available at any age. We just need to stay curious, keep learning, and enjoy the process.

6 = 3×72 - 5×42

Yes, I got the answer.

But more importantly — I saw the structure behind it.

And it was beautiful.