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?
Problem Loading...
Note Loading...
Set Loading...
Easy Math Editor
*italics*
or_italics_
**bold**
or__bold__
paragraph 1
paragraph 2
[example link](https://brilliant.org)
> This is a quote
2 \times 3
2^{34}
a_{i-1}
\frac{2}{3}
\sqrt{2}
\sum_{i=1}^3
\sin \theta
\boxed{123}
Comments
Sort by:
Top NewestYes, you just applied Euclidean algorithm.
To enter \(\LaTeX\), type like
but remove the spaces.
Log in to reply