Method 1: Lists all multiples of a number.
1. Evaluate the number to be calculated. This method is most suitable for calculating the common multiple of two numbers less than 10. If you are faced with bigger or more numbers, you'd better use other methods. For example, we need to find the least common multiple of 5 and 8. Because these two numbers are relatively small, this method is suitable for finding their least common multiple.
2. List multiples of the first number from small to large. Multiply the first number by different integers to get its multiple. In other words, you can directly look at the multiplication table and find the multiple of a number. For example, the multiple of the first number 5 is 5, 10, 15, 20, 25, 30, 35, 40.
3. Write down several multiples of the second number from small to large. Multiply the same integer by the second number to get several multiples and compare them with the previous set of multiples. In our example, the multiples of the number 8 are 8, 16, 24, 32, 40, 48, 56 and 64.
4. Compare the multiples of two numbers and find the smallest multiple. You may need to list more multiples to find the same multiple. The smallest identical number you can find is the least common multiple. For example, both multiples of 5 and 8 have 40, and they are the smallest multiples, so 40 is the least common multiple of 5 and 8.
Method 2: Use prime factor decomposition method.
1, evaluation number. This method is most suitable for calculating the common multiple of two numbers greater than 10. If you are facing a relatively small number, you'd better find the least common multiple quickly by other methods. For example, if you want to find the least common multiple of the numbers 20 and 84, you can use this method.
2. Factorize the first number. You can decompose the first number factor into its prime factor and multiply the prime factor to get the original number. You can draw a factor tree and decompose the number into prime numbers. After factorization, rewrite the equation. One side of the equation is the decomposed number, and the other side is the multiplication of prime factors. For example, 2×10 = 20 {displaystyle mathbf {2} times10 = 20}, 2× 5 =10 {displaystyle mathbf {2} times mathbf {5} =10}, so rewrite the equation and get 20 = 2× 2× 5 {displaystyle 20 = 2 imes 2 imes 5}.
3. Factorize the second number. Decompose the second number in the same way, find its prime factors and multiply each prime factor to get the second number. For example, 2× 42 = 84 {displaystyle mathbf {2} imes42 = 84}, 7× 6 = 42 {displaystyle mathbf {7} imes6 = 42}, 3× 2 = 6 {displaystyle mathbf {3} imesmathbf {2} = 6}. So the prime factors of 84 are 2, 7, 3, 2. Rewrite the equation and get 84 = 2× 7× 3× 2 {displaystyle 84 = 2× 7× 3× 2}.
4. Write down each identical prime factor and multiply each factor to write a multiplication equation. When you write down each factor, please cross out the corresponding value in the factorization equation. For example, two numbers have the same factor of 2, so write down the factor of 2×{ display style 2 times} and cross out the 2 in each factor.
Two numbers have another 2 as the same factor of * * *, so write down the second number 2 and write it as the product of two numbers: 2× 2 {displaystyle2mimes2}, and then cross out the other 2 in the factorization formula.
5. Add the remaining factors to the multiplication formula. Cofactor refers to the factors that are not crossed out after the common factors are crossed out in several factorization equations. That is, the factors of the two numbers are different. For example, in equation 20 = 2×2×5 {display style 20 = 2×2×5}, two 2' s are the same factor of two numbers * * *, so you will cross out two 2' s, leaving 5 pounds. Add 5 to the above multiplication formula to get: 2×2×5 {display style 2×2×5}.
In equation 84 = 2× 7× 3× 2 {displaystyle 84 = 2× 7× 3× 2}, you also crossed out two 2s, leaving 7 and 3. Add these two numbers to the multiplication and it becomes: 2× 2× 5× 7× 3 {display style 2× 2× 5× 7× 3}.
6. Calculate the least common multiple. Multiply all the factors written above to get the least common multiple. In our example, 2× 2× 5× 7× 3 = 420 {DisplayStyle2imes2imes5imes7imes3 = 420}. So the least common multiple of 20 and 84 is 420.
Method 3: Use grid method or trapezoid method.
1, draw a well-shaped grid. The TIC-tac-toe grid consists of two groups of parallel lines that cross each other, and the two groups of parallel lines are perpendicular to each other, forming a grid with three rows and three columns, which looks like a TIC-tac-toe key (#) on a mobile phone or keyboard. Write your first number in the top middle of the grid and your second number in the upper right corner of the grid. For example, if you want to find the least common multiple of the numbers 18 and 30, please write 18 in the top center box and 30 in the upper right corner box.
2. Find the factors that both numbers have * * *. Write the number in the box in the upper left corner of the grid. It is better to use prime factor, which will greatly facilitate the subsequent calculation, but it is not necessary. In the example of solving the least common multiple of 18 and 30, because 18 and 30 are even numbers, they can be evenly divided by 2 and written in the square in the upper left corner of the grid.
3. Divide the two numbers in the example by the same factor. Write the divided quotient in the box below each number. You can get the quotient by division. For example, 18 ÷ 2 = 9 {displaystyle18 div 2 = 9}, and write 9 under the number18.
30 ÷ 2 = 15 {displaystyle 30dv2 =15}, and write15 in the grid under 30.
4. Find the common factor of two quotients. If two quotients have no common factor, you can skip this step and go directly to the next step. If they have common factors, please write them in the box on the left of the center. For example, the common factor of 9 and 15 is 3, so write 3 in the grid on the left side of the center.
5. Divide the quotient obtained in the first step by the new common factor. Write the results below the results of the previous step. For example, 9÷3=3{displaystyle 9div 3=3}, and write 3 in the box below 9.
15÷3 = 5 {display style 15 div 3 = 5}, and write 5 in the box below 15.
6. If necessary, continue to expand the TIC-tac-toe grid and draw it bigger. Then calculate the division according to the above steps until the two quotients do not have the same factor.
7. Circle the numbers in the first column and the last row of the table. When the circles are connected, it is like drawing a capital letter "L". Multiply all the circled numbers. In our example, 2 and 3 are in the first column of the grid, and 3 and 5 are in the last row of the grid. Write the mathematical formula: 2×3×3×5 {display style 2×3×3×5}.
8. Complete the multiplication operation. Multiply all the factors and the result is the least common multiple of the first two numbers. For example: 2×3×3×5 = 90 {display style 2×3×3×5 = 90}. So the least common multiple of 18 and 30 is 90.
Method 4: Use Euclid algorithm.
1, understand the nouns in division. "Divided number" is a number divided by another number in the division operation; "Divider" is the number divided by dividend; Quotient is the final result of division; "Remainder" is the number left after an integer is divisible. For example, in the equation15 ÷ 6 = 2,3 {displaystyle15diV6 = 2; { ext { Yu } }3}:
15 is a bonus.
6 is a divisor.
2 is business.
3 is the remainder.
2. Rewrite the equation into the form of "quotient-remainder". The formula is dividend = divisor × quotient+remainder. You need to use this formula to find the greatest common divisor of two numbers according to Euclid algorithm. For example,15 = 6× 2+3 {displaystyle15 = 6 imes 2+3}.
The greatest common divisor is the greatest divisor or factor shared by two numbers.
Using this method, you need to find the greatest common divisor first, and then find the least common multiple through it.
3. Use the larger one of the two numbers as the dividend and the smaller one as the dividend. Establish the "quotient-remainder" equation of two numbers. For example, if you need the least common multiple of 2 10 and 45, the equation is in the form of 2 10 = 45×4+30 {display style 2 10 = 45 times 4+30}.
4. Take the original divisor as the new divisor and the remainder as the new divisor. Establish the "quotient-remainder" equation of two numbers. For example, 45 = 30× 2+15 {displaystyle 45 = 30 times 2+ 15}.
5. Repeat this process until the last remainder becomes 0. In each new equation, you need to use the original divisor as the new dividend and the remainder as the new divisor. For example, 30 =15× 2+0 {displaystyle30 =15 times 2+0}. Because the last remainder is 0, there is no need to continue division.
6. Find the divisor in the last equation. This number is the greatest common divisor of two numbers. For example, because the divisor in the previous equation 30 = 15x2+0 {displaystyle30 = 15times2+0} is15,15 is the greatest common divisor of 2 10 and 45.
7. Find the product of two numbers. Divide their product by their greatest common divisor. The final result is the least common multiple of two numbers. For example, 210× 45 = 9450 {displaystyle210imes45 = 9450}. Divide the product by the greatest common divisor to get 945015 = 630 {displaystyle {frac {9450} {15}} = 630}. Therefore, 630 is the least common multiple of 2 10 and 45.
It is suggested that if the least common multiple of multiple numbers is required, the above method needs to be changed slightly. For example, to find the least common multiple of16,20,32, please use the above method to find the least common multiple of16,20 (80). Then find the least common multiple of 80 and 32, and the final calculation result is 160.
The least common multiple has many uses. The most common use is that when you calculate the addition and subtraction of fractions, the denominator figures of several fractions must be the same; If the denominators are different, you need to multiply the numerator and denominator by a number at the same time, so that the denominators of several fractions become the same number. The best way is to find the least common denominator (LCD), that is, the least common multiple (LCM) of the denominator.