Creating a number using a combination of two numbers

[musicgold]

Hi,

My question is related to the following puzzle.

“What is the highest number that can’t be created by adding any number of 4s and 9s”?
For example, 25 can be created as follows: 9 + 4 + 4 + 4 + 4 =25

I know that the answer is 23. I also know that the general solution to such a problem, using the numbers X and Y is (X*Y) – X – Y, when X and Y don’t have a GCF. If they have a GCF then, any number that is not divisible by the CGF can not be made using X and Y.

I have two questions.

Q1. How can I derive this formula from scratch : (X*Y) – X – Y ?

Q2. If I am given a number 12345 to figure if it can be created using X and Y, what is the quickest way to do that?

Thanks.

 

[HallsofIvy]

(X*Y)- X- Y is NOT a formula. A formula would be saying that is equal to something.

What you mean by "created using X and Y"?

Do you mean "find the largest number, N, that cannot be written in the form "XY- X- Y= N"?

 

[Mute]

(X*Y)- X- Y is NOT a formula. A formula would be saying that is equal to something.

What you mean by "created using X and Y"?

Do you mean "find the largest number, N, that cannot be written in the form "XY- X- Y= N"?

I believe the question statement is actually, "What is the largest number, N, that cannot be made out of additions of any number of X's and Y's?", and the OP states that the answer is given by N = XY - X - Y, when X and Y have no GCF.

The OP's first question is then, how does one derive that N = XY-X-Y?

The second question is, "Given a number M and numbers X and Y, how can one figure out how to write M = aX + bY, with a and b integers, assuming a solution exists?"

Is that interpretation correct, musicgold?

 

[Mensanator]

To solve for 12345, re-arrange your formula to

(AX-M)/Y=-B

In this form, iy's a Linear Congruence, so you can use the Modular Inverse
of X&Y to find A as follows:

A = invert(X,Y)*M (mod Y) = 1*12345%4 = 1

then solve fo B: (1*9-12345)/4=-B
-3084 = -B
B = 3084


Be careful, though. You CAN actually solve f0r 23, but you get A=3,B=-1.

 

[musicgold]

The OP's first question is then, how does one derive that N = XY-X-Y?

The second question is, "Given a number M and numbers X and Y, how can one figure out how to write M = aX + bY, with a and b integers, assuming a solution exists?"

Is that interpretation correct, musicgold?

That is correct. Thanks.

 

[Mensanator]

Oh, I forgot to mentio: if you don't like A=1, pick another.
In a linear congruence, if A is a solution, so is A+Y,
or A+nY, for that matter. So we can chose any A, as
long as it's a multiple of four plus one.

For instance, we can pick A=1001 and recalculate B
(B=834), giving us: 1001*9 +834*4=12345.