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A prime number is a natural number greater than 1 whose only divisors are 1 and itself, and a composite number has at least 3 divisors. Knowing that 1 is neither prime nor composite helps you start prime factorization and divisor problems correctly.

A power is a compact way to express repeated multiplication of the same number or letter using a base and an exponent. Powers are essential for writing prime factorizations concisely, and they make calculations easier because the exponent directly shows the number of factors.

Prime factorization means expressing a natural number greater than 1 as a product of prime factors only. Apart from the order of multiplication, the result is unique, making it a starting point for finding divisors, square numbers, the greatest common divisor, and the least common multiple.

The greatest common divisor is the greatest of the common divisors of two or more natural numbers, and every common divisor is a divisor of the greatest common divisor. After prime factorization, multiply the common prime factors using the smaller exponent for each. This is often used in problems about dividing things equally.

The least common multiple is the smallest common multiple of two or more natural numbers, and all common multiples are multiples of the least common multiple. After prime factorization, multiply all the prime factors using the greater exponent for each, so it is often used in problems involving simultaneous departures.

Integers include positive integers, 0, and negative integers. Rational numbers are numbers that can be expressed as fractions whose numerators and denominators are integers. Understanding the classification of numbers helps you accurately identify integers and rational numbers that are not integers among given numbers.

A number line is a straight line with 0 as the reference point, where positive numbers are placed to the right and negative numbers to the left at equal intervals. Every rational number can be represented by a point, making it useful for visually comparing numbers and checking absolute values.

The absolute value of a number is the distance between the point representing that number on the number line and the origin. Since it is a distance, it is always greater than or equal to 0. Problems often use the fact that two numbers with the same absolute value are the same distance from the origin on opposite sides.

On a number line, numbers farther to the right are greater. A positive number is greater when its absolute value is larger, while a negative number is smaller when its absolute value is larger. Learning this together with expressions that use inequality signs to describe ranges makes it easier to solve problems that ask you to count integers satisfying given conditions.

To add two numbers with the same sign, add their absolute values and attach their common sign. To add two numbers with different signs, take the difference of their absolute values and attach the sign of the number with the greater absolute value. Subtraction can be rewritten as addition by changing the sign of the number being subtracted, so you can use the commutative and associative properties to group the calculations.

The product and quotient of two numbers with the same sign are positive, while those of two numbers with different signs are negative. Division is calculated by rewriting it as multiplication by a reciprocal. To obtain the correct value of an expression involving powers, parentheses, and the four arithmetic operations, you must follow the order of operations.

An expression using letters represents quantities whose values are unspecified with letters and expresses their relationships mathematically. Learning the rules for omitting multiplication and division signs lets you write concise expressions for application problems involving speed, concentration, and prices.

The value of an expression is the result obtained by substituting the given numbers for its letters and calculating. When substituting a negative number, you must use parentheses to avoid sign errors. The same substitution process is also used to check solutions of equations.

Learn the meanings of term, constant term, coefficient, and degree. An expression that is the sum of one or more terms is called a polynomial. A polynomial whose highest-degree term has degree 1 is called a linear expression and provides the basis for forming and solving linear equations.

Like terms have the same variables raised to the same powers. Constant terms are also like terms with one another. Only like terms can be combined into a single term by adding or subtracting their coefficients, so they are fundamental to all calculations that simplify expressions.

To add and subtract linear expressions, expand the parentheses using the distributive law, then combine like terms. If a negative number appears before the parentheses, the signs of all terms inside them change. This is where mistakes occur most often.

An equation is an equality that can be true or false depending on the value of the unknown, while an identity is an equality that is always true for any value of the unknown. Problems about conditions often use the fact that an equality is an identity if its left and right sides become identical after simplification.

An equality remains true when the same number is added to, subtracted from, or multiplied by both sides, or when both sides are divided by the same number other than 0. These properties let us rewrite an equation in the form x=(number) and also justify converting coefficients to integers.

Transposition uses the properties of equality to move a term from one side of an equation to the other with its sign changed. Collecting terms containing variables on the left and constant terms on the right lets you quickly arrange a linear equation in the form ax=b.

일차방정식은 모든 항을 좌변으로 이항해 정리했을 때 (일차식)=0 꼴이 되는 방정식이에요. 괄호를 풀고 이항해 ax=b 꼴로 만든 뒤 양변을 a로 나누면 해를 구할 수 있어요.

Applying a linear equation means representing the quantity you want to find as an unknown, forming an equation from the conditions of the problem, and solving it. At the end, always check whether the solution needs to be a natural number, as with a count or an age, and whether it fits the problem situation.

The coordinate plane consists of the x-axis and y-axis, which meet perpendicularly at the origin. A point on it is represented by an ordered pair (x-coordinate, y-coordinate). Changing the order gives a different point, so you must follow the correct order when reading and writing coordinates.

The coordinate axes divide the coordinate plane into four regions, called quadrants 1, 2, 3, and 4 counterclockwise from the upper right. Problems often test the fact that points on the coordinate axes do not belong to any quadrant.

A graph represents the relationship between two variables as ordered pairs, plotted as points or lines on the coordinate plane. School exams often include problems that ask you to interpret real situations by reading intervals of increase, decrease, and no change.

Direct proportion is a relationship in which y becomes 2 times or 3 times its value when x becomes 2 times or 3 times its value, respectively, and is expressed as y=ax. Its graph is a straight line through the origin, rising to the right if a>0 and falling to the right if a<0.

Inverse proportion is a relationship in which y becomes 1/2 and 1/3 times its original value when x becomes 2 and 3 times its original value, and it is expressed as y=a/x. Its graph consists of a pair of smooth curves that approach the coordinate axes, and the quadrants in which they lie depend on the sign of a.

Vertically opposite angles are two opposite angles among the four angles formed when two straight lines intersect at a point. Vertically opposite angles are always equal, and the sum of two adjacent angles is 180°, making these properties fundamental to finding angles formed by intersecting lines.

Corresponding angles are two angles in the same relative position, while alternate interior angles are two angles between the two lines on opposite sides of the transversal. Identifying these positions correctly lets you apply the properties of parallel lines, making it the starting point for solving angle problems.

When two parallel lines are intersected by another line, corresponding angles and alternate interior angles are equal in measure. Conversely, if corresponding angles or alternate interior angles are equal, the two lines are parallel. Angle problems involving bent lines or paper folding are all based on these properties.

In a triangle, the length of any side is always less than the sum of the lengths of the other two sides. Three line segments must satisfy this condition to form a triangle, so it is used in triangle constructions and problems about the range of possible side lengths.

Constructing a triangle means drawing it using only an unmarked straightedge and a compass. A triangle is uniquely determined when its three sides, two sides and their included angle, or one side and the angles at its endpoints are given. These three cases correspond directly to the triangle congruence criteria.

Two triangles are congruent if their three corresponding sides are equal, if two corresponding sides and the included angle are equal, or if a corresponding side and the angles at its endpoints are equal. Congruent figures have equal corresponding sides and angles, which provides a basis for finding or proving lengths and angles.

A polygon is a plane figure enclosed by several line segments, and a diagonal is a line segment joining two nonadjacent vertices. The number of diagonals in an n-gon can be found directly using a formula, which is useful for identifying a polygon from its number of diagonals.

The sum of the three interior angles of a triangle is 180°, and an exterior angle equals the sum of the two interior angles that are not adjacent to it. Using the exterior angle property, you can bring the angles in a complex figure into a single triangle and find them quickly.

An n-gon can be divided into (n-2) triangles by drawing diagonals from one vertex, so the sum of its interior angles is 180°×(n-2). For a regular polygon, dividing this value by n gives the measure of one interior angle.

At each vertex of a polygon, the interior and exterior angles add up to 180°. Subtracting the sum of the interior angles from the total gives an exterior angle sum of 360°, regardless of the number of sides. This is the quickest tool for finding the measure of one exterior angle of a regular polygon or its number of sides.

Learn the meanings of arc, chord, secant, sector, and circular segment. In a given circle, the arc length and area of a sector are directly proportional to the measure of its central angle. The fact that chord length is not directly proportional to the central angle often appears in trick questions on school exams.

For a sector with radius r and central angle x°, the arc length and area are found by multiplying the circumference and area of the circle by x/360, respectively. The area can also be found using only the radius and arc length, which also applies to calculating the lateral surface area of a cone.

A polyhedron is a solid bounded entirely by polygonal faces. Typical examples include prisms, pyramids, and frustums of pyramids. Knowing the shape of the base lets you express the numbers of faces, edges, and vertices using formulas, making counting problems easier.

A regular polyhedron is a polyhedron whose faces are all congruent regular polygons and in which the same number of faces meet at each vertex. There are only five types. They are classified by the shape of their faces and the number of faces meeting at a vertex, and they often appear in problems involving nets.

A solid of revolution is formed by rotating a plane figure through 1 revolution about a straight line. Examples include cylinders, cones, conical frustums, and spheres. Problems often test the fact that a section through a plane containing the axis of rotation is a figure with line symmetry, while a section through a perpendicular plane is a circle.

The surface area of a prism or cylinder is the sum of the areas of its two bases and its lateral surface area, and its volume is the product of its base area and height. Unfolding the lateral surface of a cylinder gives a rectangle whose width is the circumference of the base, making the surface area easy to calculate.

The surface area of a pyramid or cone is the sum of its base area and lateral surface area, and its volume is 1/3 of the volume of a prism or cylinder with the same base area and height. The lateral surface of a cone unfolds into a sector, so you can find its surface area using the formula for the area of a sector.

The surface area of a sphere of radius r is 4πr², and its volume is 4/3πr³. The volumes of a cone, a sphere, and a cylinder with the same base radius and height are in the ratio 1:2:3, so you can remember these formulas by connecting them to the volumes of cones and cylinders.

A measure of central tendency represents the characteristics of an entire data set with a single number. Examples include the mean, median, and mode. You should choose a suitable measure depending on whether the data contain extreme values and whether they are quantitative.

A stem-and-leaf plot splits data values into stems and leaves and arranges the leaves for each stem in ascending order. It lets you see the original data values as well as the shape of the distribution.

A frequency distribution table groups data into classes of equal width and shows the frequency, which is the number of data values in each class. You need to understand classes, class width, and frequency accurately to solve problems involving histograms and relative frequencies.

A histogram is a graph made by placing rectangles side by side, with the class widths from a frequency distribution table as their widths and the frequencies as their heights. Each rectangle's area is directly proportional to its frequency, so the fact that the ratio of areas equals the ratio of frequencies is often used.