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This shortcut gives the number of divisors by adding 1 to the exponent of each prime factor in the prime factorization and multiplying the results. You do not need to list every divisor, so you can quickly solve problems about the number of divisors of large numbers or finding numbers with a given number of divisors.

After prime factorization, multiply once more by each prime factor whose exponent is odd to make all exponents even. You can find the smallest number to multiply or divide by to obtain a perfect square in a single calculation.

This technique uses the relationship that the product of two natural numbers equals the product of their greatest common divisor and least common multiple. Knowing just two of the three values lets you find the remaining one with a single division, making it especially useful for problems that ask you to work backward from given conditions to find a number.

When division by several numbers gives the same remainder, use the fact that subtracting that remainder from the number you seek gives a common multiple. Simply add the remainder to the least common multiple to handle a remainder-condition problem with a single equation.

After converting two fractions to a common denominator, multiply both the denominator and numerator by the same number to increase the denominator and reveal fractions between them. This lets you easily create as many as you need in problems asking for fractions or non-integer rational numbers between two numbers.

This technique lets you read the absolute value of the difference between two numbers as the distance between two points on a number line. When working with conditions involving absolute values or finding numbers equally distant from a point, a diagram makes the answer immediately visible and reduces mistakes.

For an expression containing only multiplication and division, first determine the sign: positive if the number of negative numbers is even, and negative if it is odd. Once the sign is determined, you only need to calculate with the absolute values, which greatly reduces sign errors in products of several numbers and calculations involving powers.

This method organizes each situation in a distance–speed–time problem into a row of a table and fills in missing entries using an unknown. Once the table is complete, equal distances or times are easy to see, so you can immediately decide which quantities to equate.

In concentration problems, use the amount of salt rather than the concentration to write an equation stating that the amounts before and after mixing are equal. The amount of salt stays unchanged even when water is added or evaporated, so complex concentration problems can be expressed as a single-line linear equation.

When a solution of an equation is given, substitute that value for x to obtain an equation involving the unknown constant. A single substitution can turn problems involving equations with the same solution or equations containing constants into simple calculations.

When coefficients contain decimals or fractions, multiply both sides by a power of 10 or the least common multiple of the denominators to obtain integer coefficients. This avoids fraction calculations, shortens the calculation, and reduces sign errors when expanding parentheses.

Rewrite the equation in the form ax=b, then determine the number of solutions by checking whether a is 0 and, if a=0, whether b is 0. This lets you cover every case in problems asking for conditions for no solutions or infinitely many solutions and avoid tricky mistakes.

When the amount of work is not given, set the total work to 1 and express the amount done per day as a fraction. When several people work together or divide the work, you only need to add their daily work amounts, making it easier to set up an equation.

This technique uses the fact that the age difference between two people does not change, whether you go forward or backward in time. You can express both people's ages using just one unknown, making equations for age problems simpler.

For reflection across the x-axis, change the sign of the y-coordinate; across the y-axis, change the sign of the x-coordinate; about the origin, change the signs of both coordinates. This technique lets you write the coordinates of symmetric points immediately without drawing, making problems about shapes formed by symmetric points quicker to solve.

좌표평면 위 삼각형을 축에 평행한 직사각형으로 감싼 뒤 바깥의 직각삼각형들을 빼서 넓이를 구하는 요령이에요. 밑변과 높이가 축과 나란하지 않은 삼각형도 좌표 차이만으로 넓이를 구할 수 있어요.

This is a method for determining which quadrant a point lies in by looking only at the signs of its x-coordinate and y-coordinate. In problems asking for the location of a point with letter coordinates, first determine the signs using conditions such as ab>0 and a+b<0, and the quadrant follows immediately.

This technique reads a steeper line on a time-distance graph as a greater speed because more distance is covered in the same time. You can compare the graphs of two people or first assess changes in speed between intervals without numerical calculations.

This technique finds a by substituting the coordinates of one point on the graph into y=ax or y=a/x. For direct proportion, divide y by x; for inverse proportion, multiply x and y. You can then immediately find other points on the graph.

반비례 그래프 위의 점에서 두 축에 수선을 그어 만든 직사각형의 넓이가 항상 a의 절댓값이라는 요령이에요. 점의 위치가 달라도 넓이가 같아서 넓이 조건으로 a를 구하거나 삼각형 넓이를 구할 때 계산이 거의 없어져요.

In 1 minute, the minute hand rotates 6°, and in 1 minute, the hour hand rotates 0.5°. To find the smaller angle between the hands, take the smaller of their angular difference d calculated over a 12-hour cycle and 360°-d.

This technique draws an auxiliary line parallel to the given parallel lines at each bend, splitting angles into alternate interior or corresponding angles. The angle at a bend splits into the sum of two alternate interior angles, so the technique also applies directly to unfamiliar parallel-line configurations.

When a zigzag line lies between two parallel lines, the sum of the angles opening to the left equals the sum of those opening to the right. This rule lets you solve zigzag angle problems directly with one equation without repeatedly drawing auxiliary lines.

This technique uses the fact that folding paper makes the two angles that overlap across the fold line equal in size. In problems involving folded rectangular paper, combining this with alternate interior angles reveals isosceles triangles and makes hidden angles easier to find.

To determine whether three line segments can form a triangle, this technique checks only whether the longest side is shorter than the sum of the other two sides. There is no need to check all three inequalities, so you can quickly find the range even when the lengths contain variables.

This technique chooses the appropriate congruence criterion from SSS, SAS, and ASA by examining the positions of the sides and angles given as equal. You only need to check whether the angle lies between the two sides or the side lies between the two angles, so you can quickly find the basis for a congruence proof.

From one vertex of an n-gon, draw n-3 diagonals, excluding the vertex itself and its two neighbors, then divide the total counted twice by 2. This method also applies to inverse problems that ask you to identify a polygon from its number of diagonals and to counting problems involving handshakes or round-robin tournaments.

Use the fact that a triangle's exterior angle equals the sum of its two nonadjacent interior angles to combine a star's five tip angles within one triangle. Their sum is always 180°, allowing you to solve star angle problems directly.

The exterior angles of a polygon always sum to 360°, so each exterior angle of a regular n-sided polygon is 360°/n. This calculation is simpler than the interior angle sum formula, making it much faster to identify a regular polygon from an interior or exterior angle.

A sector's arc length and area are the circle's circumference and area multiplied by the central angle's proportion. Using their direct proportionality to the central angle, a single proportion lets you find an unknown arc length, area, or central angle.

This method finds a sector's area using only its radius and arc length, without the central angle. As with triangle area, multiply the radius by the arc length and divide by two, which also shortens calculations of a cone's lateral surface area.

For a convex polyhedron, subtracting the number of edges from the number of vertices and adding the number of faces gives 2. For a polyhedron satisfying this condition, you can use two of the three counts to find the remaining count or check your calculations.

Unfold the solid into a net, view its lateral surfaces as rectangles or sectors, and add their areas to those of the bases. A cylinder's lateral surface becomes a rectangle whose width is the base circumference, and a cone's becomes a sector, allowing you to find the entire surface area.

First find the volume of a prism or cylinder with the same base area and height, then divide it by 3 to find the volume of the pyramid or cone. The same principle applies to both pyramids and cones, so volumes of cut solids or frustums can also be calculated easily by subtracting pyramid or cone volumes from prism or cylinder volumes.

When there is an even number of data values, this technique orders them by size and takes the mean of the two middle values as the median. In median problems involving unknown values, identifying the two middle values first reduces mistakes.

When data contain extremely large or small values, use the median instead of the mean as the representative value. Extreme values greatly affect the mean, but the median hardly changes, so it better represents the center of the data.

In a stem-and-leaf plot, the leaves are sorted by size, so you can determine ranks by counting leaves from the top or bottom. Without listing the data again, you can directly find the value at a given rank from the largest or the number of values at or above a particular value.

Find an erased frequency in a frequency table or histogram by subtracting the known frequencies from the total frequency. If a ratio condition is also given, assign variables to the missing frequencies and form a linear equation to find both values together.

This technique uses the fact that the sum of the rectangle areas in a histogram equals the class width multiplied by the sum of the frequencies. The area enclosed by the frequency polygon and the horizontal axis is also the same, so area problems can be solved with a single multiplication.

When the frequency and relative frequency of a class are known, recover the total frequency by dividing the frequency by the relative frequency. Once you have the total frequency, you can find the frequencies of all other classes even from a torn table or a partially visible graph.

This technique converts frequencies to relative frequencies before comparing two groups with different total frequencies. It lets you fairly compare proportions or distributions in two classes with different numbers of students and avoid pitfalls in graph interpretation problems.

Reduce the fraction to lowest terms, factor its denominator into primes, and check only whether there are prime factors other than 2 and 5. You can immediately tell whether the decimal terminates without dividing, and this is especially useful in problems asking for the smallest number to multiply by.

To convert a repeating decimal to a fraction, write 9 in the denominator for each digit in the repeating block and 0 for each non-repeating digit after the decimal point. This lets you convert directly to a fraction without setting up and subtracting equations, saving time and reducing errors.

밑이 달라 보이는 거듭제곱을 4를 2의 제곱으로 보는 식으로 같은 밑으로 바꾸는 요령이에요. 밑이 같아지면 지수끼리만 비교하면 되므로 미지수가 지수에 있는 식도 쉽게 풀려요.

In a product of powers of 2 and powers of 5, pair 2 with 5 to form powers of 10. Looking only at the remaining number and the exponent of 10 lets you count the digits of a large number without calculating it.

This technique uses the fact that the units digits of powers repeat in a fixed cycle. The remainder when the exponent is divided by the cycle length lets you immediately find the units digit of even a very large power.

When evaluating an expression, first combine like terms to simplify it instead of substituting numbers immediately. Having fewer terms to substitute into greatly reduces mistakes in calculations with fractions or negative numbers.

어떤 식에 A를 빼야 할 것을 잘못 더한 결과가 주어지면, 그 결과에서 A를 빼 원래 식부터 되찾는 요령이에요. 원래 식을 구한 뒤 바르게 계산하면 되어 문제 구조가 단순해져요.