Melody

A two-digit number has the property that the square of its tens digit plus ten times its units digit equals the square of its units digit plus ten times its tens digit. Determine all two-digit numbers which have this property, and are prime numbers.

let the number be   10a+b  where a and b are single digit integers and a is not 0

the square of its tens digit  (a^2)  plus ten times its units digit (10b) equals the square of its units digit (b^2) plus ten times its tens digit (10a)

\(a^2+10b=b^2+10a\\ a^2-10a=b^2-10b\)

If you cannot see the the answer off hand then try looking at a= 1 up to a =9  one by one and see if you can find a b value that will work.

Thanks Coolstuff, I like your explanation.    

Since you are interested in teaching I thought you might be interested in alternate techniques .

This is how I would do it if I was serioulsy teaching someone about expanding brackets.

I'd talk about the area of a rectangle.  So a rectangle with sides 2 and 3 would have an area of  2*3 = 6

I'd draw that

Then I would move onto an algebraic example.  I would start with one that only had plus signs but I have drawn a representation of the first half of this problem.  

3(4x-2y) 

So 

    \(3(4x-2y)\\=12x+-6y\\=12x-6y\)

I have not studied these answers but I doubt very much that either one is rubbish.

Tiggsy and Heureka are two of our top mathematicians here. 

Thanks for your answers guys. I am hoping I find time to study them.    

Fractions are really hard for people when they are first learning them!

I think any totally new concept can be really hard. Once you have the basis for a topic then it gets much easier.

It does for me anyway.

So if you are very little division might seem impossible.    

Thanks

Yes maybe I could do it in spreadsheet if I wanted to but I was really trying to extract the logic.

When you answer using a program it is nice if you say that so everyone knows where your answers comes from. 

I like people doing this. I am not trying to discourage you at all. 

I'll give you an outline:

Use similar triangles to show that TY is congruent to TW

Now you have three congruent sides and one pair is parallel.

Now draw in the diagonal YW 

You now have enough info to show that triangles  TYW  and XYW are congruent.

Now you can prove the rest.

Hi guest.

How did you work all these totals out?

Thanks Heureka,

Am I right in thinking you just used a formula.

I mean can you see the logic behind the forumula - I can't.