Math Is Fun Forum

Welcome!

The linear Diophantine equation ax + by = c (where a, b, c are integers) has a solution if and only if c is a multiple of the greatest common divisor (aka the highest common factor) of a and b.

For example, 15x + 12y = 500 has no solutions because the GCD (or HCF) of 15 and 12 is 3, which doesn't divide 500.
And 17x + 29y = 1000 must have a solution in the integers because the GCD (or HCF) of 17 and 29 is 1, which divides 1000.

Also, once you've found a solution (x, y), then all the other solutions have the form

for an arbitrary integer k.

You can extend this trick to find integer solutions to much more complicated equations (e.g. quadratics, cubics) but you'd need to do a bit more work. For example, you can take modulo 4 of both sides of the equation

to show that this doesn't have any integer solutions.

Welcome!

Looks like ChatGPT was used to produce the content of Alexander_Thomas's post -- probably with navyboy484's post entered as a prompt. That writing style is unmistakable. Ever since the software was released, bots are using it to try to create more convincing/human-like responses.

For example, if I type the following into ChatGPT:

navyboy484 wrote:
Just an easy problem for you guys to solve (I don’t need help with this just wanted to post something easy here to pass the time).

Solve for “x”

2.1x = 23.1

...then ChatGPT 3.5 gives me this:

Sure, I'd be happy to solve the equation for "x"!

The equation is:
2.1x = 23.1

To solve for "x," we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 2.1:

(2.1x) / 2.1 = 23.1 / 2.1

The 2.1 on the left side cancels out, leaving us with:

x = 23.1 / 2.1

Now, let's calculate the value of "x":

x = 11

So, the solution to the equation is x = 11.

Not exactly the same as post #3, but we can see that the format is very similar.

Welcome!

Welcome!

Hi Jack Omar,

The question we want to answer is: how do we describe the 'slope' of a function at any given point on its graph? Well, as the pictures you've added show, we can

(a) Take the first point on the curve, find another point on the curve, then draw a straight line through them...
(b) ...then calculate the slope of the line you've just drawn.

Looks OK to me -- isn't the last line the same as the first one?

I saw that you asked some questions about integration as well -- but let me know if this makes sense first and then we can move on.

Welcome!

Zazu, it's perfectly OK if you'd like to share a link to another maths resource and there is a dedicated subforum -- Maths Teaching Resources -- for doing just that (and we will be happy to support you in doing so).

What isn't OK is for you to pretend to ask for help, return to your post a day later and edit it to insert a hidden link to your website. It gives off the impression that you are using our forum as nothing more than a bulletin board to redirect traffic to your site. Please stop overriding my edits -- if you keep doing so I will have no choice but to restrict your access to the forum.

Just a precautionary note -- while Bob's approach in post #6 is correct, we can only do this because the series converges. In other words, the logic would be:

(1) First prove that the series converges, e.g. by looking at the sequence of partial sums (that's the sum of the first N terms) and showing that that sequence converges as N gets larger.
(2) Now that we know the sum converges, we can then do the rearranging of terms that Bob outlined in post #6 to identify what it converges to.

If the series did not converge then this kind of manipulation is not allowed -- take for example the divergent series 1 - 1 + 1 - 1 + 1 - ... which can be rearranged to 'converge' to lots of different things.

Similar reasoning is also required for differentiating/integrating the infinite series in post #2. We can do this here because p is a probability (so is between 0 and 1, within the series' radius of convergence) and so the series is integrable within that radius of convergence (and its integral also converges).

If we know the square has area 33cm^2 then we don't need to know anything about the side length to determine its perimeter -- was the question intended to be 'find x'?