Determine the positive integers

In summary, positive integers are whole numbers greater than 0 that do not include fractions or decimals. To determine if a number is a positive integer, it must be greater than 0 and a whole number. The main difference between positive integers and negative integers is the sign, with positive integers represented by a plus sign and negative integers represented by a minus sign. Not all whole numbers are positive integers, as whole numbers include 0, which is neither positive nor negative. Positive integers are important in mathematics as they form the basis for many mathematical concepts and are used in various equations and formulas to solve problems.
  • #1
anemone
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Determine the positive integers $z>y>x$ for which

$\dfrac{1}{x}-\dfrac{1}{xy}-\dfrac{1}{xyz}=\dfrac{19}{97}$
 
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  • #2
anemone said:
Determine the positive integers $z>y>x$ for which

$\dfrac{1}{x}-\dfrac{1}{xy}-\dfrac{1}{xyz}=\dfrac{19}{97}$

(1/x)-(1/xy)-(1/xyz)=19/97
=> 1- 1/y – 1/yz = 19x/97
Or 1/y + 1/yz = (97-19x)/97 so 19 x < 97 or x <=5
Now as 97 is in denominator on RHS so z has to be multiple of 97
If the value of z is chosen to be < 97 for simplicity and no solution with higher z can be proved

x = 1 => 1/y + 1/yz = 78/97 no solution as y has to be 2 and 1/2 + 1/6 = 2/3 < 78/97
x = 2 => 1/y + 1/yz = 59/97 no solution as 1/3 + 1/12 = 5/12 < 59/97
x= 3 => 1/y+ /yz = 40/97 no solution as 1/4 + 1/20 = 3/10 < 40/97
x= 4 => 1/y + 1/yz = 21/97 no solution as 1/5 + 1/(5* 97) = 98/(5 * 97) < 21/97
x =5 => 1/y + 1/yz = 2/97
or 97z + 97 = 2yz
or y = 97(z+1)/(2z)
if y = 97m then m = (z+1)/2z. m > 1 is not possible and m = 1 => z = 1 but z > y so no solution

z has to odd and multiple of 97
z= 97 => y = 49
z = 97 *3 => y = 146 and no solution for z > 97
so z = 97 and y = 48 is solution
so x = 5, y = 48 and z = 97 is the solution
 

FAQ: Determine the positive integers

What are positive integers?

Positive integers are whole numbers that are greater than 0. They do not include any fractions or decimals.

How do you determine if a number is a positive integer?

To determine if a number is a positive integer, you can check if it is greater than 0 and if it is a whole number. If both conditions are met, then the number is a positive integer.

What is the difference between positive integers and negative integers?

The main difference between positive and negative integers is the sign. Positive integers are represented by numbers with a plus sign (+), while negative integers are represented by numbers with a minus sign (-). Additionally, positive integers are greater than 0, while negative integers are less than 0.

Are all whole numbers positive integers?

No, not all whole numbers are positive integers. Whole numbers include 0, which is neither positive nor negative. Positive integers only include numbers that are greater than 0.

Why are positive integers important in mathematics?

Positive integers are important in mathematics because they form the basis for many mathematical concepts, such as addition, subtraction, multiplication, and division. They are also used in various mathematical equations and formulas to solve problems and find solutions.

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