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?

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
`2 \times 3`	\( 2 \times 3 \)
`2^{34}`	\( 2^{34} \)
`a_{i-1}`	\( a_{i-1} \)
`\frac{2}{3}`	\( \frac{2}{3} \)
`\sqrt{2}`	\( \sqrt{2} \)
`\sum_{i=1}^3`	\( \sum_{i=1}^3 \)
`\sin \theta`	\( \sin \theta \)
`\boxed{123}`	\( \boxed{123} \)

Excel in math and science