The main property of the fraction, formulation, proof, examples of application. The main property of the fraci: the wording, proof, examples of the application as the main property of the fraction

In this article we will analyze what is the main property of the fraction, we formulate it, we give the proof and a visual example. Then we consider how to apply the main property of the fraction when performing actions to reduce fractions and bring fractions to a new denominator.

All ordinary fractions have an essential property that we call the main property of the fraction, and it sounds as follows:

Definition 1.

If the numerator and denominator of one fraction is multiplied or divided into one and the same natural number, then the event will result in a fraction equal to the specified one.

Imagine the basic property of the fraction in the form of equality. For natural numbers a, b and m, equality will be fair:

a · m b · m \u003d a b and A: M B: M \u003d A B

Consider the proof of the main properties of the fraction. Based on the properties of multiplying the natural numbers and the properties of the division of natural numbers, we write the equality: (a · m) · b \u003d (b · m) · a and (a: m) · b \u003d (b: m) · a. Thus, fractions a · m b · m and A B, as well as a: M b: M and A B are equal to the definition of equality of fractions.

We will analyze an example that graphically illustrates the main property of the fraction.

Example 1.

Suppose we have a square divided into 9 "large" portions. Each "big" square is divided into 4 smaller in size. It is possible to say that the specified square is divided into 4 · 9 \u003d 36 "small" squares. We highlight the color of 5 "big" squares. At the same time, there will be 4 · 5 \u003d 20 "small" squares. Let's show the drawing showing our actions:

The painted part is 5 9 source figures or 20 36, which is the same. Thus, fractions 5 9 and 20 36 are equal: 5 9 \u003d 20 36 or 20 36 = 5 9 .

These equality, as well as equality 20 \u003d 4 · 5, 36 \u003d 4 · 9, 20: 4 \u003d 5 and 36: 4 \u003d 9 make it possible to conclude that 5 9 \u003d 5 · 4 9 · 4 and 20 36 \u003d 20 · 4 36 · 4.

To consolidate the theory, we will analyze the solution of the example.

Example 2.

It is specified that the numerator and the denominator of some ordinary fraction was multiplied by 47, after which these numerator and the denominator were divided into 3. Is the fraction given as a result of these actions?

Decision

Based on the main property of the fraction, we can say that the multiplication of the numerator and the denominator of the given fraction on the natural number 47 will result in a fraction equal to the source. We can argue the same, producing further division by 3. Ultimately, we will get a fraction equal to the specified.

Answer: Yes, the resulting fraction will be equal to the initial one.

Application of the main properties of the fraction

The main property is applied when it is necessary to bring the fraction to a new denominator and with a reduction of fractions.

Bringing fractions to a new denominator is the action of the replacement of a given fraction equal to it, but with a large numerator and denominator. To bring a fraction to a new denominator, you need to multiply the numerator and denominator of the fraction on the necessary natural number. Actions with ordinary fractions would be impossible without a way to bring a fraction to a new denominator.

Definition 2.

Reduction of fractions - The transition to a new fraction equal to a given, but with a smaller numerator and denominator. To shorten the fraction, you need to divide the numerator and denominator of the fraction on the same natural number that will be called common divider.

There may be cases when there is no such common divider, then they suggest that the initial fraction is inconspicuous or not subject to reduction. In particular, the reduction of the fraction with the help of the greatest common divisor will lead to a fraction of an incomprehensive mind.

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Fraction - form of representation of a number in mathematics. The fractional feature indicates the operation of the division. Numerator The fraci is divisible, and denominator - Divider. For example, the fraction of the numerator is the number 5, and the denominator is 7.

Right It is called a fraction that has a numerator module larger than the denominator module. If the fraction is correct, then its module is always less than 1. All other fractions are wrong.

Fraction is called mixedif it is recorded as an integer and fraction. This is the same as the amount of this number and fractions:

The main property of the fraci

If the numerator and denominator of the fraction are multiplied by the same number, then the value of the fraction will not change, that is, for example,

Bringing fractions to a common denominator

To bring two fractions to a common denominator, you need:

1. The numerator of the first fraction multiply to the denominator of the second
2. Numerator of the second fraction multiplying the denominator
3. Rannels of both fractions replace their work

Actions with fractions

Addition. To fold two fractions, you need

1. Folded new numerals of both fractions, and the denominator is left unchanged

Example:

Subtraction. To subtract one fraction from another, you need

1. Bring a fraction to a common denominator
2. Subtract from the numerator of the first fraction Numerator second, and the denominator is left unchanged

Example:

Multiplication. To multiply one fraction to another, multiply their numerators and denominators.

Shares of one and appears in the form \\ FRAC (A) (B).

Slipter fraction (A) - The number above the fraction feature and showing the number of shares to which the unit was divided.

Dannel of fractions (b) - The number under the feature of the fraction and showing how many fractions shared a unit.

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The main property of the fraci

If ad \u003d bc, then two fractions \\ FRAC (A) (B)and \\ FRAC (C) (D) are considered equal. For example, the fractions will be equal \\ FRAC35and \\ FRAC (9) (15)since 3 \\ Cdot 15 \u003d 15 \\ CDOT 9, \\ FRAC (12) (7)and \\ FRAC (24) (14)Since 12 \\ Cdot 14 \u003d 7 \\ CDOT 24.

From the definition of equal fractions it follows that the fractions will be equal \\ FRAC (A) (B)and \\ FRAC (AM) (BM)Since A (Bm) \u003d B (AM) is a visual example of the use of combinating and moving properties of multiplying natural numbers in action.

So \\ FRAC (A) (B) \u003d \\ FRAC (AM) (BM) - It looks like the main property of the fraci.

In other words, we will get a fraction equal to this, multiplying or separating the numerator and denominator of the initial fraction on the same natural number.

Reduction of fractions - This is the process of replacing the fraction, in which a new fraction is obtained equal to the original, but with a smaller numerator and denominator.

Reducing the fraction is made, based on the main property of the fraction.

For example, \\ FRAC (45) (60) \u003d \\ FRAC (15) (20)(Numerator and denominator is divided into number 3); The resulting fraction can be reduced again, dividing on 5, that is \\ FRAC (15) (20) \u003d \\ FRAC 34.

Unstable fraction - it fraction like \\ FRAC 34.where the numerator and the denominator are mutually simple numbers. The main goal of the cutting of the fraction is to do a fraction disrox.

Bringing fractions to a common denominator

Take two fractions as an example: \\ FRAC (2) (3)and \\ FRAC (5) (8) with different denominators 3 and 8. In order to bring these fractions to a common denominator and first change the numerator and denominator \\ FRAC (2) (3)on 8. We get the following result: \\ FRAC (2 \\ CDOT 8) (3 \\ CDOT 8) \u003d \\ FRAC (16) (24). Then multiply the numerator and denominator \\ FRAC (5) (8)by 3. We get in the end: \\ FRAC (5 \\ CDOT 3) (8 \\ CDOT 3) \u003d \\ FRAC (15) (24). So, the initial fractions are given to the total denominator 24.

Arithmetic actions on ordinary fractions

Addition of ordinary fractions

a) With the same denominators, the numerator of the first fraction is folded with the Nizer of the second fraction, leaving the denominator for the same. As can be seen in the example:

\\ FRAC (A) (B) + \\ FRAC (C) (B) \u003d \\ FRAC (A + C) (B);

b) With different denominators, the fractions first lead to a common denominator, and then perform the addition of numerals according to rule a):

\\ FRAC (7) (3) + \\ FRAC (1) (4) \u003d \\ FRAC (7 \\ CDOT 4) (3) + \\ FRAC (1 \\ CDOT 3) (4) \u003d \\ FRAC (28) (12) + \\ FRAC (3) (12) \u003d \\ FRAC (31) (12).

Subtracting ordinary fractions

a) with the same denominators from the numerator of the first fraction, the Numerator of the second fraction is subtracted, leaving the denominator for the same:

\\ FRAC (A) (B) - \\ FRAC (C) (B) \u003d \\ FRAC (A-C) (B);

b) If the denominators of fractions are different, then first the fractions lead to a common denominator, and then repeat the actions as in paragraph A).

Multiplication of ordinary fractions

Multiplication of fractions obeys the following rule:

\\ FRAC (A) (B) \\ CDOT \\ FRAC (C) (D) \u003d \\ FRAC (A \\ CDOT C) (B \\ Cdot D),

that is, separately numerals and denominers.

For example:

\\ FRAC (3) (5) \\ CDOT \\ FRAC (4) (8) \u003d \\ FRAC (3 \\ CDOT 4) (5 \\ CDOT 8) \u003d \\ FRAC (12) (40).

Division of ordinary fractions

Division fractions produce in the following way:

\\ FRAC (A) (B): \\ FRAC (C) (D) \u003d \\ FRAC (AD) (BC),

that is a fraction \\ FRAC (A) (B) multiplied by fraction \\ FRAC (D) (C).

Example: \\ FRAC (7) (2): \\ FRAC (1) (8) \u003d \\ FRAC (7) (2) \\ CDOT \\ FRAC (8) (1) \u003d \\ FRAC (7 \\ CDOT 8) (2 \\ Cdot 1 ) \u003d \\ FRAC (56) (2).

Mutually reverse numbers

If AB \u003d 1, then the number B is in return For number a.

Example: for number 9 reverse is \\ FRAC (1) (9), as 9 \\ CDOT \\ FRAC (1) (9) \u003d 1for number 5 - \\ FRAC (1) (5), as 5 \\ CDOT \\ FRAC (1) (5) \u003d 1.

Decimal fractions

Decimal fraction It is called the correct fraction, the denominator of which is 10, 1000, 10 \\, 000, ..., 10 ^ n.

For example: \\ FRAC (6) (10) \u003d 0.6; \\ ENSPACE \\ FRAC (44) (1000) \u003d 0.044.

In the same way, it is written incorrect with a denominator 10 ^ n or mixed numbers.

For example: 5 \\ FRAC (1) (10) \u003d 5.1; \\ ENSPACE \\ FRAC (763) (100) \u003d 7 \\ FRAC (63) (100) \u003d 7.63.

In the form of a decimal fraction, any ordinary fraction with a denominator is represented, which is a divider of some number 10.

Example: 5 - divider of the number 100, so the fraction \\ FRAC (1) (5) \u003d \\ FRAC (1 \\ CDOT 20) (5 \\ CDot 20) \u003d \\ FRAC (20) (100) \u003d 0.2.

Arithmetic actions over decimal fractions

Addition of decimal fractions

To add two decimal fractions, it is necessary to position them so that each other is the same discharges and the comma dilated, and then make the fraction of fractions as ordinary numbers.

Subtraction decimal fractions

Performed similar to the addition.

Multiplying decimal fractions

Upon multiplying decimal numbers, it is sufficient to multiply the specified numbers, not paying attention to the commas (as natural numbers), and in the resulting answer of the comma on the right, so many digits are separated, as they are after the semicolon in both factors total.

Let's perform a multiplication of 2.7 per 1.3. We have 27 \\ Cdot 13 \u003d 351. We separate the right two digits of the semicolon (at the first and second number - one digit after the comma; 1 + 1 \u003d 2). As a result, we get 2.7 \\ Cdot 1,3 \u003d 3.51.

If in the resulting result, there are fewer numbers than it is necessary to separate the comma, then the missing zeros are written ahead, for example:

For multiplication by 10, 100, 1000, it is necessary in the decimal fraction to transfer the comma to 1, 2, 3 digits to the right (if necessary, a certain number of zeros is attributed to the right).

For example: 1.47 \\ CDOT 10 \\, 000 \u003d 14,700.

Division of decimal fractions

The division of the decimal fraction on a natural number is also produced as the division of a natural number on natural. The comma in private is placed after the division of the whole part is completed.

If a whole part of the divisible less divider, it turns out to be zero as well, for example:

Consider the division of the decimal fraction on the decimal. Let it be necessary to divide 2.576 per 1.12. First of all, smartly dividimit and divider of fractions 100, that is, we will transfer the comma to the right in Delima and the divider to so many signs as they are in the divider after a comma (in this example for two). Then it is necessary to perform the division of the fraction 257.6 to the natural number 112, that is, the task is reduced to the case already considered:

It happens that the ultimate decimal fraction is not always obtained when dividing one number to another. As a result, an infinite decimal fraction is obtained. In such cases, transfers to ordinary fractions.

2.8: 0.09 \u003d \\ FRAC (28) (10): \\ FRAC (9) (100) \u003d \\ FRAC (28 \\ CDOT 100) (10 \\ CDOT 9) \u003d \\ FRAC (280) (9) \u003d 31 \\ FRAC (1) (9).

This topic is sufficiently important on the basic properties of fractions, all further mathematics and algebra are based. The considered properties of fractions, despite its importance, very simple.

To understand the main properties of fractions Consider a circle.

On the circle it can be seen that 4 parts or painted from eight possible. We write the resulting fraction \\ (\\ FRAC (4) (8) \\)

On the next circle, it can be seen that one part of two possible is painted. We write down the fraction \\ (\\ FRAC (1) (2) \\)

If you look closely, we will see that in the first case, in the second case, we have half a circle, so the resulting fractions are equal to \\ (\\ FRAC (4) (8) \u003d \\ FRAC (1) (2) \\), that is This is the same number.

How to prove it mathematically? Very simple, remember the multiplication table and with the first fraction on multipliers.

\\ (\\ FRAC (4) (8) \u003d \\ FRAC (1 \\ CDOT \\ Color (Red) (4)) (2 \\ CDOT \\ COLOR (RED) (4)) \u003d \\ FRAC (1) (2) \\ CDOT \\ Color (Red) (\\ FRAC (4) (4)) \u003d \\ FRAC (1) (2) \\ CDOT \\ COLOR (RED) (1) \u003d \\ FRAC (1) (2) \\)

What did we do? Signed a numerator and denominator for multipliers \\ (\\ FRAC (1 \\ CDOT \\ COLOR (RED) (4)) (2 \\ CDOT \\ COLOR (RED) (4)) \\), and then divided the fractions \\ (\\ FRAC (1) (2) \\ CDOT \\ Color (Red) (\\ FRAC (4) (4)) \\). Four divided into four this is 1, and the unit multiplied to any number is this itself. What we did in the example called reducing fractions.

Let's see another example and reduce the fraction.

\\ (\\ FRAC (6) (10) \u003d \\ FRAC (3 \\ CDOT \\ COLOR (RED) (2)) (5 \u200b\u200b\\ CDOT \\ Color (Red) (2)) \u003d \\ FRAC (3) (5) \\ CDOT \\ Color (Red) (\\ FRAC (2) (2)) \u003d \\ FRAC (3) (5) \\ CDOT \\ COLOR (RED) (1) \u003d \\ FRAC (3) (5) \\)

We again painted the numerator and denominator for multipliers and the same number in the numerals and denominators have shown. That is, two divided into two gave a unit, and the unit multiplied to any number gives the same number.

The main property of the fraction.

Hence the main property of the fraci:

If the numerator, and the denominator of the fraci multiply the same number (except zero), then the fraction will not change.

\\ (\\ BF \\ FRAC (A) (B) \u003d \\ FRAC (A \\ CDOT N) (B \\ Cdot N) \\)

You can also fly the numerator and denominator to share the number at the same time.
Consider an example:

\\ (\\ FRAC (6) (8) \u003d \\ FRAC (6 \\ DIV \\ Color (Red) (2)) (8 \\ DIV \\ Color (Red) (2)) \u003d \\ FRAC (3) (4) \\)

If the numerator, and the denomote denoter to share the number (except zero), then the size of the fraction will not change.

\\ (\\ BF \\ FRAC (A) (B) \u003d \\ FRAC (A \\ DIV N) (B \\ Div N) \\)

The fractions of which are in the numerals, and in the denominants, common ordinary dividers are called social fraud.

Example of reduced fraction: \\ (\\ FRAC (2) (4), \\ FRAC (6) (10), \\ FRAC (9) (15), \\ FRAC (10) (5), ... \\)

There is also unstable fractions.

Unstable fraction - This is a fraction of which there are no numbers and denominators of common ordinary divisors.

Example of an inconspicuous fraction: \\ (\\ FRAC (1) (2), \\ FRAC (3) (5), \\ FRAC (5) (7), \\ FRAC (13) (5), ... \\)

Any number can be represented as a fraction, because any number is divided by one, eg:

\\ (7 \u003d \\ FRAC (7) (1) \\)

Questions to the topic:
What do you think anyone can shorten or not?
Answer: No, there are reduced fractions and non-interpretable fractions.

Check whether the equality is true: \\ (\\ FRAC (7) (11) \u003d \\ FRAC (14) (22) \\)?
Answer: Scroll fraction \\ (\\ FRAC (14) (22) \u003d \\ FRAC (7 \\ CDOT 2) (11 \\ Cdot 2) \u003d \\ FRAC (7) (11) \\)Yes, rightly.

Example number 1:
a) Find the fraction with the denominator 15 equal to the fraction \\ (\\ FRAC (2) (3) \\).
b) Find a fraction with a numerator 8 equal to the fraction \\ (\\ FRAC (1) (5) \\).

Decision:
a) we need the number 15 in the denominator. Now in the denominator number 3. What number needs to multiply the number 3 to get 15? Recall the multiplication table 3⋅5. We need to take advantage of the main property of fractions and multiply and numerator, and denominator \\ (\\ FRAC (2) (3) \\)by 5.

\\ (\\ FRAC (2) (3) \u003d \\ FRAC (2 \\ CDOT 5) (3 \\ CDOT 5) \u003d \\ FRAC (10) (15) \\)

b) We need a number 8 in the numerator 8. Now there are numbers in numerators 1. What number do you need to multiply the number 1 to get 8? Of course, 1⋅8. We need to take advantage of the main property of fractions and multiply and numerator, and denominator \\ (\\ FRAC (1) (5) \\) On 8. We will get:

\\ (\\ FRAC (1) (5) \u003d \\ FRAC (1 \\ CDOT 8) (5 \\ CDOT 8) \u003d \\ FRAC (8) (40) \\)

Example number 2:
Find an inconspicuous fraction, equal to the fraction: a) \\ (\\ FRAC (16) (36) \\),b) \\ (\\ FRAC (10) (25) \\).

Decision:
but) \\ (\\ FRAC (16) (36) \u003d \\ FRAC (4 \\ CDOT 4) (9 \\ CDOT 4) \u003d \\ FRAC (4) (9) \\)

b) \\ (\\ FRAC (10) (25) \u003d \\ FRAC (2 \\ CDOT 5) (5 \\ CDOT 5) \u003d \\ FRAC (2) (5) \\)

Example number 3:
Write down the number in the form of a fraction: a) 13 b) 123

Decision:
but) \\ (13 \u003d \\ FRAC (13) (1) \\)

b) \\ (123 \u003d \\ FRAC (123) (1) \\)

From the course algebra of the school program proceed to specific. In this article, we will examine the special type of rational expressions in detail - rational fractionsand also we will analyze what characteristic identical transformation of rational fractions take place.

Immediately note that rational fractions in the sense in which we will define them below, in some textbooks, algebra is called algebraic fractions. That is, in this article we will understand the same thing under rational and algebraic fractions.

Let us begin with definition and examples. Next, let's talk about bringing a rational fraction to a new denominator and about the change of signs in the members of the fraction. After that, we will analyze how the frains are reduced. Finally, we will focus on the representation of a rational fraction in the form of a sum of several fractions. All information will be supplied with examples with detailed descriptions of solutions.

Navigating page.

Definition and examples of rational fractions

Rational frarators are studied in the lessons of algebra in grade 8. We will use the definition of rational fraction, which is given in the textbook of algebra for 8 classes Yu. N. Makarychev, etc.

In this definition, it is not specified whether polynomials in the numerator and denominator of the rational fraction should be polynomials of the standard form or not. Therefore, we assume that in the records of rational fractions can be found both polynomials of the standard species and not standard.

We give a few examples of rational fractions. So, X / 8 and - rational fractions. And the fraci And they are not suitable for the voiced definition of rational fraction, since in the first of them in the numerator it is not a polynomial, but in the second and in the numerator and in the denominator are expressions that are not polynomials.

Transformation of the numerator and denominator of rational fraction

The numerator and denominator of any fraction are self-sufficient mathematical expressions, in the case of rational fractions, these are polynomials, in the particular case - are unoccupied and numbers. Therefore, with a numerator and denominator of rational fraction, as with any expression, can be carried out identical conversions. In other words, the expression in the rational fraction numerator can be replaced by an identical expression equal to it, as well as the denominator.

In the numerator and denominator of rational fraction, identical conversions can be performed. For example, in the numerator you can carry out a grouping and bringing similar terms, and in the denominator - the product of several numbers replace it with a value. And since the numerator and denominator of rational fraction are polynomials, then with them you can also perform and characteristic of the polynomials of the transformation, for example, bringing to a standard form or representation in the form of a piece.

For clarity, consider solutions to several examples.

Example.

Convert rational fraction So that the polynomial is a polynomial of a standard species in the numerator, and in the denominator - the product of polynomials.

Decision.

The creation of rational fractions to a new denominator is mainly used when adding and subtracting rational fractions.

Changing signs before fraction, as well as in its numeric and denominator

The main property of the fraction can be used to change the signs from the members of the fraction. Indeed, the multiplication of the numerator and the denominator of the rational fraction on -1 is equivalent to the change of their signs, and the result is a fraction, identically equal to this. It is often necessary to contact this transformation when working with rational fractions.

Thus, if you simultaneously change the signs in the numerator and denominator of the fraction, it will turn out the fraction equal to the original one. Equality is responsible for this statement.

Let us give an example. The rational fraction can be replaced by identically equal to the fraction with the changed signs of the numerator and the denominator of the species.

With fractions, one more identical conversion can be carried out at which the sign changes either in the numerator or in the denominator. Let's voice the appropriate rule. If you replace the fraction sign together with the number of the number or denominator, it will turn out to the fraction, identically equal to the source. Recorded statement correspond to equality and.

Proving these equality is not difficult. The proof is based on the multiplication properties of numbers. We prove the first of them :. With the help of similar transformations, equality is proved.

For example, the fraction can be replaced by expression or.

In conclusion of this paragraph, we give two more useful equality and. That is, if you change the sign only in the numerator or only by the denominator, the fraction will change its sign. For example, and .

Considered transformations that allow you to change the sign in the members of the fraction, often apply when converting fractional rational expressions.

Reducing rational fractions

At the heart of the following transformation of rational fractions having a name reduction of rational fractions, is also the main property of the fraction. This transformation corresponds to equality where A, B and C are some polynomials, and B and C - nonzero.

From the given equality it becomes clear that the reduction of the rational fraction involves the disposal of the total factor in its numerator and the denominator.

Example.

Reduce the rational fraction.

Decision.

A general multiplier 2 is visible, we will perform a reduction on it (when recording, general factors that are reduced, convenient to cross out). Have . Since x 2 \u003d x · x and y 7 \u003d y 3 · y 4 (see if necessary), it is clear that X is a common multiplier of the numerator and denominator of the resulting fraction, like Y 3. We will reduce these factors: . This reduced reduction.

Above, we have reduced the rational fraction consistently. And it was possible to reduce the reduction in one step, immediately reducing the fraction by 2 · x · y 3. In this case, the solution would look like this: .

Answer:

.

With a reduction in rational fractions, the main problem is that the total multiplier of the numerator and the denominator is not always visible. Moreover, it does not always exist. In order to find a common factor or make sure that it is not necessary for a numerator and denominator of rational fraction to decompose on multipliers. If there is no common factor, then the initial rational fraction does not need a reduction, otherwise there is a reduction.

In the process of reduction of rational fractions, various nuances may occur. The main subtleties on the examples and in the details disassembled in the article reducing algebraic fractions.

Completing the conversation about the reduction of rational fractions, we note that this transformation is identical, and the main complexity in its conduct is to decompose the polynomials in the numerator and denominator.

Representation of rational fraction in the form of the amount of fractions

Quite specific, but in some cases very useful, it turns out to transform a rational fraction, which consists in its representation as a sum of several fractions, or the sum of the whole expression and fraction.

The rational fraction, in the numerator of which there is a polynomial, which is a sum of several universions, can always be written as the amount of fractions with the same denominators, in whose numerators are appropriate. For example, . Such a submission is explained by the rule of addition and subtracting algebraic fractions with the same denominators.

In general, any rational fraction can be represented as a fraction by a variety of different ways. For example, the fraction A / B can be represented as the sum of two fractions - arbitrary fractions C / D and fraction, equal difference fractions A / B and C / D. This statement is fair, as there is equality . For example, a rational fraction can be represented as a sum of fractions in various ways: Imagine the initial fraction in the form of the sum of the whole expression and the fraction. After dividing the numerator to the denominator, we will get equality . The value of the expression N 3 +4 for any whole n is an integer. And the fraction value is an integer then and only if its denominator is 1, -1, 3 or -3. These values \u200b\u200bcorrespond to n \u003d 3, n \u003d 1, n \u003d 5 and n \u003d -1, respectively.

Answer:

−1 , 1 , 3 , 5 .

Bibliography.

• Algebra: studies. For 8 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorov]; Ed. S. A. Telikovsky. - 16th ed. - M.: Enlightenment, 2008. - 271 p. : IL. - ISBN 978-5-09-019243-9.
• Mordkovich A. G. Algebra. 7th grade. In 2 tsp. 1. Tutorial for students of general educational institutions / A. Mordkovich. - 13th ed., Act. - M.: Mnemozina, 2009. - 160 p.: Il. ISBN 978-5-346-01198-9.
• Mordkovich A. G. Algebra. 8th grade. In 2 tsp. 1. Tutorial for students of general educational institutions / A. Mordkovich. - 11th ed., Ched. - M.: Mnemozina, 2009. - 215 p.: Il. ISBN 978-5-346-01155-2.
• Gusev V. A., Mordkovich A. G. Mathematics (benefit for applicants in technical schools): studies. benefit. - m.; Higher. Shk., 1984.-351 p., Il.