How do I use the approximate and least common multiples?

Whether you know it or not, let me tell you a few properties and inferences:

I won't talk about multiples, divisors, divisibility and the like.

1.The product of the greatest common divisor and the least common multiple of the two positive integers a and b is equal to the product of these two numbers.

For example, if the greatest common divisor and least common multiple of a natural number and 24 are 4 and 168, respectively, find this natural number.

It's very simple: Use property 1 to find the answer.

The corollary of it can be obtained from 1: if (a,b)=1 then [a,b]ab

It's also very proven. There's another nature that you may not understand, but I'll type it out first:

2.Let the positive integers a > b, and a=bq+r (r less than or equal to b-1 greater than or equal to 0), where q and r are integers, then (a,b)=(b,r).

This means that two numbers treat a large number as a small number of b times plus the integer r

Then the greatest common divisor of a,b is equal to the greatest common divisor of b,r.

There are two questions: two two-digit numbers, the greatest common divisor is 8, the least common multiple is 96, and the sum of these two numbers is?

There are several apples, two in a pile with one more, three in a pile with one more, four in a pile with one more, five in a pile with one more, six in a pile with one more, how many apples are in this pile?

All in all, the use of the greatest common divisor and the least common multiple is flexible and changeable, and you need to be flexible when doing the problem to win all battles.

I wish you more and more success in your studies

To give you a debut question:

a,b)=c

a,b]=d

then a*b=?

a, b) denotes the ab greatest common divisor.

a, b] denotes the greatest common multiple of ab.

To do the question, these two symbols are clarified first.

There are two methods for calculating common multiples, namely the decomposition prime factor method and the formula method, and the specific methods are:

1. Decomposition of prime factor method

Write out the prime factors of these numbers first, and the least common multiple is equal to the product of all their prime factors (if there are several prime factors that are the same, compare which of the two numbers has the more prime factors and multiplies more times).

For example, find the least common multiple of 45 and 30.

The greatest common divisor, the least common multiple.

The different prime factors are that 3 is the prime factor that they both have, and since 45 has two 3s and 30 has only one 3, multiply two 3s when calculating the least common multiple. The least common multiple is equal to 2*3*3*5=90

Another example is to calculate the least common multiple of 36 and 270 = 2*3*3*3*5

The different prime factors are that this prime factor is more in 36, which is two, so multiply it twice; The prime factor of 3 is more than 270, which is three, so it is multiplied by three times. The least common multiple equal to and 40 is 40.

2. Formula method

Since the product of two numbers is equal to the product of the greatest common divisor and the least common multiple of these two numbers. i.e. (a,b) [a,b]=a b. So, to find the least common multiple of two numbers, you can first find their greatest common divisor, and then use the above formula to find their least common multiple.

For example, if you find [18,20], you get [18,20]=18 20 (18,20)=18 20 2=180. To find the least common multiple of several natural numbers, you can first find the least common multiple of two of the numbers, and then find the least common multiple of this least common multiple and the least common multiple of the third number, and then continue until the last one. The resulting least common multiple is the least common multiple of the numbers sought.

The least common multiple is 3x5x1x3x1 45, as shown in the following figure:

The short division sign is the division sign reversed. Short division is to write the prime factor common to two numbers where the divisor is written in division, and then drop the quotient of the two numbers divisible by the common prime factor, and then divide, and so on, until the result is coprime (the two numbers are co-primed).

When calculating the common multiple number by short division, the factor of the existence of any two of the sock numbers must be calculated, and the other numbers without this factor will fall as they are. Until every two remaining are mutual.

To find the greatest common divisor is to multiply by one side, and to find the least common multiple to multiply by one circle.

Take finding the least common multiple of , for example.

Find out the least common factor of first, column short division.

Remove these numbers with the least common factor of 2, which is the sum of three numbers, to get three quotients.

Find the least common factor 2 of the three quotients, and remove these quotients with the least common factor to get a new level of quotient.

And so on, until the final quotient is co-qualified (i.e., several quotients have only a common factor of 1).

Multiply all the common factors and the resulting quotients, and the resulting product is the least common multiple of the numbers we require.

As shown below: <>

Hope to have your ex-brother help.

It is common practice to write each of these numbers as the product of prime numbers, e.g. to calculate the least common multiple.

The same parts of the equation are then combined into a single prime number, which is then multiplied by the remaining prime numbers.

There are no identical parts in the above three formulas, only 3, 2*2, and 5 are not the same parts.

So the least common multiple is 3*2*2*5=60.

Similarly, the least common multiple of 6,12,18 is found as follows:

You see, the three 2s in the above three formulas are merged into a 2, and the three 3s are merged into a 3, leaving a 2 and a 3, and the multiplication of the slider is 2*3*2*3=36.

Step 1: Find the least common factor of two numbers, column the short division, and use the least common factor to remove these two numbers to get two quotients;

Step 2: Then find out the least common factor of the two quotients, and remove the two quotients with the least common factor to get the two quotients of the new level;

Step 3: And so on until the two quotients are coprime numbers (i.e., the two quotients have only a common factor of 1);

Step 4: Multiply all the common factors and the last two quotients, and the product is the least common multiple of the two numbers we require.

Example 1: Find the least common multiple of 3, 12, 20.

1) Find the greatest common divisor of 3 and 12 3

2) Find the greatest common divisor of 4 and 20.

3) Multiply each factor by 3 4 1 1 5 = 60

Example 2: Find the least common multiple of 36,100,105.

1) Find the greatest common divisor of 36 and 100 4

2) Find the greatest common divisor of 25 and 105 5

3) Find the greatest common divisor of 9 and 21 3

4) Multiply each factor by 4 5 3 3 5 7 = 6300

In addition, it is also possible to find the least common multiple by factoring the prime factor.

In Example 1: 3=3 1,12=2 2 3,20=2 2*5

Because the highest power of 2 is 2, the highest power of 3 is 1, and the highest power of 5 is 1, the least common multiple is 2 2 3 5 = 60

In example 2: 36 = 2 2 3 2,100 = 2 2 5 2,105 = 3 5 7

Because the highest power of 2 is 2, the highest power of 3 is the exciter 2, the highest power of 5 is 2, and the highest power of 7 is 1, so the least common multiple is 2 2 * 3 2 5 2 * 7 = 6300

Greatest common factor.

Decomposition of prime factors: It is to decompose several numbers into the form of prime factors, and multiply the common factors to obtain the maximum common factor.

(12, 18).

Find the least common multiple.

To find the least common multiple of several numbers, the commonly used methods are:

1) Find the least common multiple of several numbers, first see if there is a common divisor of these hailstones (not necessarily the common divisor of all known numbers, the common divisor of any two numbers can also be), if so, use their common divisor to divide continuously, until every two numbers are coprime, and then multiply all the divisors and the final quotient, and the product is the least common multiple of these numbers.

Example: Find the least common multiple of 12 and 18. Material.

2 and 3 are coprime and so on.

The least common multiple of 12 and 18 is.

Find the least common multiple of .

Every two numbers are coprimes, except for that.

The least common multiple of is .

72。(2) Find the greatest common divisor first.

To find the least common multiple of two numbers, you can use the relationship between these two numbers and their greatest common divisor and least common multiple.

The relationship is: greatest common divisor Least common multiple = product of the multiplication of two numbers.

Example: Find the least common multiple of 12 and 18.

Solution: Because the greatest common divisor of 12 and 18 is 6, and the product of the two numbers is 12 18 216, the least common multiple of 12 and 18 is: 216 6 = 36.

3) Direct observation.

The relationship between two numbers is multiple

If the larger number is a multiple of the smaller number, then the larger number is the least common multiple of the two numbers. For example, 96 is a multiple of 16 and 96 is the least common multiple of 96 and 16.

The two numbers are coprimitive:

If two numbers are coprime, then the least common multiple of the two numbers is the product of the two numbers. Example: The least common multiple of 7 and 13 is.

For example, find the most common common factor of 12 and 18.

The factor of 12 has .

The factor of 18 has .

The common factor between 12 and 18 is .

The greatest common factor between 12 and 18 is 6.

This method is obviously inconvenient for finding the greatest common factor of more than two numbers, especially for larger numbers. Therefore, the method of decomposing the prime factors for each number was adopted.

12 and 18 can be divided into several different forms of products, but divided into prime factor products only one of the above, and can no longer be decomposed. The divided prime factors are undoubtedly divisible by the originals, so they are also divisors of the originals. From the results of decomposition, 12 and 18 both have common factors 2 and 3, and their product 2 3 6 is the greatest common factor of 12 and 18.

The method of decomposing prime factors is also in the form of short division, but it is divided separately, and then the common factor and the greatest common factor are found. It's easier if you combine these two numbers together and divide them.

It is not difficult to see from the short division that 12 and 18 both have common factors 2 and 3, and their product 2 3 6 is the greatest common factor of 12 and 18. Compared with the previous decomposition of the prime factors, it can be found that not only the results are the same, but also the left side of the short division vertical is the common prime factor of these two numbers, and the greatest common factor of the two numbers is the product of the common prime factors of these two numbers.

In practical application, two or more numbers that need to be calculated are placed together for short division, as shown in Figure 1.

When calculating the least common multiple of multiple numbers, the factors that exist in any two of them are calculated, and the other numbers that do not have this factor are left as they are. Finally, multiply all the factors with the last two remaining numbers that are coprime (there is no other common factor except 1) to get the least common multiple. See Figure 2.

The least common multiple is required to find the greatest common divisor first. There is only one common divisor of 7 and 4 or 10, and the common divisor of 4 and 10 is 2, and then 7 * (10 2) * (4 2) = 140, because 10 and 4 have a common divisor 2, so the least common multiple is 140 as shown in the figure.