Euclidean algorithm

I found something interesting about solving this question:

If p is the number of pairs of values (x, y) that can make the equation 84x + 140y = 28 true, what is true about p?

I noticed while solving that I could simplify the equation:

84x - 140y = 28

42x – 70y = 14

21x – 35y = 7

3x – 5y = 1

Noting that 3x = 5y + 1 means that 3x must end in 5 + 1 = 6 or 0 + 1 = 1 to allow y to be divisible by 5.

This means 3x should end in either 1 or 6 to be equal to 5y + 1

Finding the following values for x and matching values for y:

3x = 6, 21, 36, 51, 66, 81, 96, 111, 126, 141, 156

5y = 5, 20, 35, 50, 65, 80, 95, 110, 125, 140, 155

Then finding the series of x and y:

X = 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52

Y = 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31

X seems to be increasing by 5, and Y by 3.

Which means that you can find pairs of x and y by the formulas:

x = 2+5a and y = 1+3a

Leading to a sequence that will satisfy 84x + 140y = 28.

Would this also be more generally true? I also thought it might be an interesting additional question to the quiz.

On another note, this text editor does not seem to like enters in paragraphs. Any way I can fix this?

#NumberTheory

Note by Sytse Durkstra
3 years, 8 months ago

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Comments

Yes, you just applied Euclidean algorithm.

To enter LaTeX\LaTeX, type like

\ ( ... \ )

but remove the spaces.

Pi Han Goh - 3 years, 8 months ago
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