Many thanks. Thinking out aloud: 10x+12y =4 - why not simplify to 5x + 6y =2 And see concept of complementary solution ax+by = 0 and particular solution (x=-2, y= 2) .
If a and b are relatively prime positive integers, prove that the Diophantine equation ax - by = c has infinitely many solutions in the positive integers. [Hint: There exist integers xo and Yo such that axo + byo = c. For any integer t, which is larger than both I xo I / b and I Yo I /a, a positive solution of the given equation is x = xo + bt, y =-(yo - at).]
I usually solve this using modular arithmetic 10x+12y=4 5x+6y =2 6y=2-5x ( mod 5) y=2 (mod 5) y=5k +2 Sub in the equation 5x + 6(5k+2) =2 5x +30k +12 =2 5x=-30k-10 x=-6k-2 This is the same answer as tge video with replacing k with -k
what if you are given something like 63x - 23y = -7. The gcd (63, -23) = 1 and 1 | -7. but do you solve the equation 63x - 23y = 1 and then multiply by -7? I'm faced with this problem and confused.
Yes, solve for 63x - 23y = 1, it will gives value x and y both positive, then general solutions will be (x, y)= {(x0 - bn), (y0 - an). Then multiply by -7 to get the final solution.
His explanation covers this. 0 is a multiple of the gcd, so solutions will exist. In fact, they wouldn't be hard to find. You won't even need to go through the gcd route. For example, 10x + 12y = 0, then x = -6y/5. To make x an integer, choose y = 5k, where k is an integer, and then you get x = -6k. So all integer solutions have the form (x,y) = (-6k, 5k), for any integer k.
Michael Penn, I want to be a mathematician like you. Super smart and great explaining but also fit and hot. Thank you, sir.
Professor Penn , thank you for a powerful explanation of the Linear Diophantine Equations in classic Number Theory.
Great Explanation Sir
Diophantine? More like dio-phantastic! Thanks again for making and posting these wonderful videos!
Many thanks. Thinking out aloud: 10x+12y =4 - why not simplify to 5x + 6y =2
And see concept of complementary solution ax+by = 0 and particular solution (x=-2, y= 2) .
Wow, that is beautiful and thoroughly explained.
Clearly i understand sir ... Great explanation .
Thanks for the videos professor michael,
Greetings from colombia 🍃
Thank you so much for taking the time to provide this education
If a and b are relatively prime positive integers, prove that the Diophantine equation
ax - by = c has infinitely many solutions in the positive integers.
[Hint: There exist integers xo and Yo such that axo + byo = c. For any integer t,
which is larger than both I xo I / b and I Yo I /a, a positive solution of the given equation is
x = xo + bt, y =-(yo - at).]
Great video 👍
good explanation sir
yeah but what if GCD = 1, or a & b are relatively prime?
Then there is a solution for every integer c!
Wow
use Euclidean Algorithm， can find x，y s.t
ax+by=1 than
C(ax+by=1)
The bezout idenity is in a later video but this says 'recall' which threw me off a lot
I usually solve this using modular arithmetic
10x+12y=4
5x+6y =2
6y=2-5x ( mod 5)
y=2 (mod 5)
y=5k +2
Sub in the equation
5x + 6(5k+2) =2
5x +30k +12 =2
5x=-30k-10
x=-6k-2
This is the same answer as tge video with replacing k with -k
elegant!
Nodular Asthmatics lmao
@@utkarshsharma9563 spelling mistake
@@skwbusaidi I thought you wrote that as a joke
Where did the mod 5 come from?
Can we do it the other way every such equation represents a straight line and we need to find number of lattice points the straight line hits??
But it will take time to do it
The method is right
sir solution for 11x+y=11 please
god u saved me thank u so muchhhhhhhhhhhhhhhhhhhhhhhh
Sir, Do you upload group theory lessons?
It would be very helpful!
He has alot of that stuff under his Abstract Algebra playlist I believe. Hope that helps
what if you are given something like 63x - 23y = -7. The gcd (63, -23) = 1 and 1 | -7. but do you solve the equation 63x - 23y = 1 and then multiply by -7? I'm faced with this problem and confused.
Yes, solve for 63x - 23y = 1, it will gives value x and y both positive, then general solutions will be (x, y)= {(x0 - bn), (y0 - an).
Then multiply by -7 to get the final solution.
More on diophantine equations : czcams.com/video/6rjoO4K_XuI/video.html
from korea i have a Q
What is diopantin 423x + 198y = 24 ?
Plz teach me
gcd(423,198) = 9, and 24 is NOT a multiple of 9. Therefore, this equation has no solution.
what if , 35x + 21y = 1 ?
This is not solvable because gcd(21,35)=7.
Recall x and y are integers only in our scenario
I think c=0 needs to be explain also
His explanation covers this. 0 is a multiple of the gcd, so solutions will exist. In fact, they wouldn't be hard to find. You won't even need to go through the gcd route. For example, 10x + 12y = 0, then x = -6y/5. To make x an integer, choose y = 5k, where k is an integer, and then you get x = -6k. So all integer solutions have the form (x,y) = (-6k, 5k), for any integer k.
It is given that a,b,c are elements of Natural numbers or Positive integers 0:01. c cannot be equal to 0
@@clawjet6069 c = 0, means ax+by=0, is 2nd type equation, where ax+by=c, is 1st type equation.
"And that's a good place to stop" ?????
Yes
where is the backflip penn!!!!!!!!!!!!!!!!!!!!?
Clearly i understand sir ... Great explanation .