Integer values of this expression

In summary, to find the integer values of f(x)=(a+5x)/(6x+1) with a, x integers, you can start by choosing any integer for x, and then solving for a using the equation a = (6x + 1)n - 5x.
  • #1
epsi00
84
0
Is there a simple way to find the integer values of f(x)=(a+5x)/(6x+1) with a, x integers.
 
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  • #2
If you pick any integer for x, you can set
[tex]a = (6x + 1) n - 5x[/tex]
to get any integer n you like.

Is that what you meant by "finding the integer values"?
 
  • #3
CompuChip said:
If you pick any integer for x, you can set
[tex]a = (6x + 1) n - 5x[/tex]
to get any integer n you like.

Is that what you meant by "finding the integer values"?

In fact, it's exactly the opposite I want. a is a given integer and I am looking for the value(s) of x that make f(x) take integer values. It's simple enough for small values of a but for large values, I cannot afford to check the values of x one by one.
 
  • #4
One thing that may help, other than the obvious observation that x=0 and x=a-1 are always solutions, is that solutions (like these trivial two) come in pairs: that is, if [itex]x_0[/itex] is a solution, producing an integer [itex]n_0 = \frac {5x_0 + a} {6x_0 + 1} [/itex], then it's not hard to prove that [itex]x_1=n_0 - 1[/itex] is a solution too (which will produce the integer [itex]n_1=x_0+1[/itex]).

At least this cuts the search up to something on the order of the sqrt(a): now you need to iterate, from k=1 to a maximum of k=[itex]\lfloor \frac {2 + \sqrt{4+6a}} 6 \rfloor[/itex], checking if [itex]a \equiv k+1 \pmod {6k+1}[/itex], in which case both x=k and x=[itex]\left( \frac {5k+a} {6k+1} - 1 \right)[/itex] are two new solutions (or one, if these two "dual" solutions happen to be the same).
 
  • #5
OK, then just reverse my argument:
If n = (a + 5x) / (6x + 1), then you can solve x from this equation. That should give you something like x = (n - a) / (5 - 6n).
This will give integer values for x when (n - a) is a multiple of (5 - 6n), from which you will get the allowed values of a that will give you f(x) = n for your favorite integer n.
 

1. What are integer values?

Integer values are whole numbers, either positive or negative, that do not have any decimal places. They can be represented on a number line and include numbers such as 1, -5, 0, and 100.

2. How do you determine the integer values of an expression?

To determine the integer values of an expression, you must first simplify the expression using the order of operations. Then, you can plug in different integer values for the variables and evaluate the expression to see which values result in whole numbers.

3. What is the difference between an integer and a decimal?

The main difference between an integer and a decimal is that an integer is a whole number, while a decimal has a fractional part. Integers do not have any decimal places, while decimals can have any number of decimal places.

4. Can an expression have more than one integer value?

Yes, an expression can have more than one integer value. Depending on the complexity of the expression, there may be multiple integer values that satisfy the equation. It is important to check all possible integer values to ensure accuracy.

5. How are integer values used in real life?

Integer values are used in many real-life situations, such as counting objects, keeping track of money, and measuring distance. They are also used in mathematical equations and computer programming to represent whole numbers.

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