How to know the smallest multiples of numbers.
Miracle
To understand how to calculate the NOC, the following is determined by the meaning of the term “multiples”.
A multiple of A is a natural number that is easily divisible by A. Thus, multiples of 5 can be used for 15, 20, 25, and so on.
The divisions of a specific number can be demarcated by number, and the axis by multiples.
More divisible than natural numbers is a number that can be divided by them without excess.
How to know the smallest multiples of numbers
The smallest natural multiple (LCD) of numbers (two, three or more) is the smallest natural number that is divisible by all numbers.
To know the NOC, you can use a number of methods.
For small numbers, you can manually write down multiples of numbers in a series of dots, but there is no clue in the middle of them.
Multiples are designated by the capital letter Do.
For example, multiples of 4 can be written like this:
To (4) = (8,12, 16, 20, 24, ...)
To (6) = (12, 18, 24, ...)
So, you can note that the smallest multiples of the numbers 4 and 6 are the number 24. This entry should be concluded in the following order:
LCM(4, 6) = 24
Since the numbers are large, it is better to know the exact multiples of three or more numbers, so it is better to use another method of calculating the NOC.
For the calculation it is necessary to divide the pronunciation of numbers into simple multipliers.
You need to write the first number in the row of the largest number, and below it - the next number.
The unfolded skin number may have a variety of multiples.
For example, we can decompose the numbers 50 and 20 into simple multipliers.
In the layout of the smallest number, add the multipliers that are daily in the layout of the first largest number, and then add to the next one.
The pointed butt does not have doubles.
Now you can virahuvati in smaller multiples of 20 and 50.
LCM(20, 50) = 2 * 5 * 5 * 2 = 100
36 = 2 * 2 * 3 * 3
24 = 2 * 2 * 2 * 3
16 = 2 * 2 * 2 * 2
Thus, the addition of simple multipliers of a larger number and multipliers of a different number, which did not increase to the distribution of the larger one, will be the lowest multiple.
In this manner, they need to be added before the larger number is laid out.
LCM(12, 16, 36) = 2 * 2 * 3 * 3 * 2 * 2 = 9
There are differences in the meaning of the smallest multiple.
So, if one of the numbers can be divided without excess by another, then more of those numbers will be the least divisible.
For example, the NOC of twelve and twenty chotirioh will be twenty chotiri.
It is necessary to know the smallest multiples of mutually prime numbers, which are the basis of the new businessmen, their counterparts to their work.
For example, LCM (10, 11) = 110.
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Captions before slides:
Math lesson for 6th grade.
Mathematics teacher DBOU ZOSH No. 539 Dmitro Vadimovich Labzin.
The smallest number of times.
Sleeping robot.
Given numbers: 15 and 30. Multiples of 15: 15;
thirty; 45;
60; 75;
90... Multiples of 30: 30;
60;
90... The smallest multiple: 30. Tse tsikavo!
Multiples of 30: 30;
60;
90... The dermal multiple of the number LOC (a; b) is a multiple of the numbers a and b and, however, the dermal multiple of the number LOC (a; b).
Topic: “The smallest number of times”, 6th grade, UMK Vilenkin N.Ya.
Lesson type
: “discovery” of new knowledge
Main goals.
Determine the value of the smallest multiple of the algorithm for determining the NOC.
Formulate the procedure before finding the NOC.
Trenuvati building
Before you can understand the simple and storage number;
Authenticity mark on 2, 3, 5, 9, 10:
Various ways to find NOC:
Algorithms for finding the cross-section and combining the multiplicities;
3) It’s difficult to arrange the layout into simple multipliers.
I Self-esteem to activity.
Let's do a warm-up.
Children break into groups to find options.
Be the first to take a card from the mission and vote to your group:
1st - authenticity mark on 2;
The other is a sign of authenticity at 3;
third - authenticity mark on 5;
Fourth - authenticity mark on 9;
5th - authenticity mark on 10;
6th - authenticity mark on 2..
The numbers appear on the presentation screen: 51, 22, 37, 191, 163, 88, 47, 133, 152, 202, 403, 75, 507, 609, 708, and children are responsible for writing down those numbers. designated (or rise from the place, since until the number can be set to the sign given to it)
Guys, do you still need to know the signs of authenticity?
(For dividing numbers into multiples)
II.
So, formulate the topic of today’s lesson (in the smallest number possible)
What's the meta lesson?
(learn to know the NOC)
We found the LOC using the selection method, and by what method can we find the LOC?
(method of laying out simple multipliers)
What is the essence of this?
IV.
Get the project out of the way
Together, the children develop an algorithm for finding the LOC.
For this need:
LCM(18, 24) = 24 * 3 = 72
V. Primary consolidation in external promotion.
Worker zoshit, stor. 28 No. 3 abc
The task is completed with comments according to the output algorithm behind the proposed scheme.
VI.
Independent work with self-checking behind the eye
Learn to design independently No. 181 (abvg).
Verified correctly
Amends are corrected, their reasons are revealed and promoted.
At this hour, studies, which have been faithfully discovered, can be supplemented by No. 183
VII.
- Inclusion in the system of knowledge and repetition
Scientists, who have begun to pay attention to an independent robot, at this stage, will conclude No. 4 RT (worker sewing, page 29) to restart the smallest multiple.
Other studies are in groups No. 193, 161, 192
The captains represent the decision.
VIII.
Reflection of activity.
(Lesson result).
How can a number be called a different multiple of numbers?
What number is called the smallest multiple of numbers?
How to know the smallest multiple?
Learn to place a figure in the section from 0 to 1 that represents the level of understanding of new topics, for example
IX.
Homework.
P.7 side 29-30, No. 202, 204, 206 (ab) dodatkovo (behind the basket) No. 209 with a presentation at the upcoming lesson.
Let’s continue with Rozmova about the smallest fold, as we were told in the section “NOK - the smallest fold, chosen, applied.”
In this topic we will look at ways to find the LCM for three numbers and more, and we will look at how to find the LCM of a negative number. (126 , 70) = 14 .
The LCM is computable: LCD (126, 70) = 126 70: GCD (126, 70) = 126 70: 14 = 630.
Subject: LCM(126, 70) = 630.
Butt 2
Find the number 68 and 34.
P.7 side 29-30, No. 202, 204, 206 (ab) dodatkovo (behind the basket) No. 209 with a presentation at the upcoming lesson.
It’s hard to find a gcd, since the fragment 68 is divided by 34.
Subject: Calculable by the smallest multiple of the formula: GCD (68, 34) = 68 34: GCD (68, 34) = 68 34: 34 = 68.
LCM(68, 34) = 68.
In some applications, the rule of finding the smallest multiple of positive numbers a and b was used: if the first number is divisible by another, so that the LCM of these numbers is equal to the first number.
Knowing the NOC for additional decomposition of numbers into simple multipliers
Now let's look at the method of finding the LCM, which is based on the decomposition of numbers into simple multipliers.
Vicennia 2
- To find the smallest literal multiple, we need a few simple steps:
- we add up the sum of all prime factors of numbers for which we need to know the LCM;
- we turn off their removal of all simple multipliers;
removals after turning off the hidden prime multipliers of the original LCM of these numbers.
This is a method of finding the smallest literal multiple base on the level of GCD (a, b) = a · b: GCD (a, b).
As soon as you marvel at the formula, you become aware: the addition of numbers a and b is the same as the addition of all multipliers, which take part in the decomposition of these two numbers. With this, the GCD of two numbers is equal to the addition of all simple multipliers that are simultaneously present in multiplier decompositions of these two numbers.і Butt 3 We have two numbers 75 and 210. We can split them into multipliers like this:.
75 = 3 5 5 210 = 2 3 5 7.
If you add up the multipliers of the two output numbers, you get:
2 3 3 5 5 5 7 441 і 700 To turn off the hidden factors for both numbers, multipliers 3 and 5, we take away the solid form:
P.7 side 29-30, No. 202, 204, 206 (ab) dodatkovo (behind the basket) No. 209 with a presentation at the upcoming lesson.
2 3 5 5 7 = 1050
441 147 49 7 1 3 3 7 7
700 350 175 35 7 1 2 2 5 5 7
.
This tvir will be our LCM for the numbers 75 and 210. Butt 4 Find the LCM of numbers , breaking down the offended numbers into simple multipliers We know all the simple multipliers of numbers given for the minds: We can subtract two small numbers: 441 = 3 · 3 · 7 · 7 and 700 = 2 · 2 · 5 · 5 · 7 ..
Subject: The income of all the multipliers that took part in the distribution of these numbers, we see:
2 2 3 3 5 5 7 7 7
.
Previously, we turned off a number of multipliers for both numbers.
- Now we will do it differently:
- Let's break down the numbers into simple multipliers:
- I will add to the creation of simple multipliers of the first number and daily multipliers of the other number;
We remove the solid that will be the search for the LOC of two numbers.
Butt 5 With this, the GCD of two numbers is equal to the addition of all simple multipliers that are simultaneously present in multiplier decompositions of these two numbers.і Butt 3 Let's turn to the numbers 75 and 210, for which we have already looked for the NOC in one of the front butts. 5 Let's break them down into simple multipliers: 2 і 7 . Until the creation of multipliers 3, 5 numbers 75 added daily multipliers
numbers 210.
Ignorable:
P.7 side 29-30, No. 202, 204, 206 (ab) dodatkovo (behind the basket) No. 209 with a presentation at the upcoming lesson.
2 · 3 · 5 · 5 · 7 . This is the LCM of the numbers 75 and 210.і Butt 6 It is necessary to calculate the LCM of the numbers 84 and 648. 7
Let’s decompose the numbers from our minds into simple multipliers:
3
84 = 2 2 3 7 648 = 2 2 2 3 3 3 3.
Subject: Add to the creation of multiples 2, 2, 3
numbers 84 multipliers 2, 3, 3
numbers 648.
We remove TV
2 2 2 3 3 3 3 7 = 4536. This is the smallest multiple of the numbers 84 and 648. LCM(84, 648) = 4536. The value of the NOC of three or more numbers Regardless of how many numbers we have on the right, the algorithm of our actions will always be the same: we will consistently find the LCM of two numbers.
This is why there is a theorem.
Theorem 1
It is acceptable that we have whole numbers 250 .
P.7 side 29-30, No. 202, 204, 206 (ab) dodatkovo (behind the basket) No. 209 with a presentation at the upcoming lesson.
a 1 , a 2 , … , a k
.
NOC
m k
These numbers change during sequential calculation: m 2 = LCM (a 1, a 2), m 3 = LCM (m 2, a 3), …, m k = LCM (m k − 1, a k).
Subject: Now let's look at how you can formulate a theorem based on specific specifications.
Butt 7
It is necessary to calculate the smallest multiple of four numbers 140, 9, 54 and
Enter the values: a 1 = 140, a 2 = 9, a 3 = 54, a 4 = 250.
- Consider that m 2 = LCM (a 1, a 2) = LCM (140, 9) is computable.
- The Euclidean algorithm for calculating the GCD of the numbers 140 and 9 is simple: 140 = 9 15 + 5, 9 = 5 1 + 4, 5 = 4 1 + 1, 4 = 1 4. Reduced: GCD (140, 9) = 1, GCD (140, 9) = 140 · 9: GCD (140, 9) = 140 · 9: 1 = 1260. Otzhe, m 2 = 1260.
- before the creation finished at the advanced stage, daily multipliers of the third number are added, etc.;
- otrimany tvir will be the smallest multiple of all numbers from the mind.
Butt 8
It is necessary to know the LCM of five numbers: 84, 6, 48, 7, 143.
P.7 side 29-30, No. 202, 204, 206 (ab) dodatkovo (behind the basket) No. 209 with a presentation at the upcoming lesson.
Let's decompose all five numbers into simple multipliers: 84 = 2 2 3 7, 6 = 2 3, 48 = 2 2 2 2 3, 7, 143 = 11 13.
Simple numbers, such as the number 7, cannot be decomposed into simple multipliers.
Such numbers are avoided in their calculations for simple multipliers.
Subject: Now let's take the addition of simple factors 2, 2, 3 and 7 of the number 84 and add to them the factors of another number.
We divided the number 6 into 2 and 3.
These multipliers are already in place on the first day.
Well, let's omit them.
We can continue to add daily multipliers.
Let's go to 48, for the sake of simple multipliers we take 2 and 2. Then we add a simple multiplier of 7 on the fourth and multipliers of 11 and 13 on the fifth.і Reduced: 2 · 2 · 2 · 2 · 3 · 7 · 11 · 13 = 48048. This is the smallest multiplicity of five output numbers.
LCM(84, 6, 48, 7, 143) = 48,048. Then we add a simple multiplier of 7 on the fourth and multipliers of 11 and 13 on the fifth. The meaning of the smallest multiple of negative numbers Reduced: 2 · 2 · 2 · 2 · 3 · 7 · 11 · 13 = 48048..
In order to know the lowest multiples of negative numbers, the numbers must first be replaced with numbers with a protagonist sign, and then the calculations can be carried out using advanced algorithms.
Butt 9 − 145 і − 45 .
P.7 side 29-30, No. 202, 204, 206 (ab) dodatkovo (behind the basket) No. 209 with a presentation at the upcoming lesson.
LCM(54, -34) = LCM(54, 34), and LCM(-622, -46, -54, -888) = LCM(622, 46, 54, 888). − 145 і − 45 Such actions are permissible in connection with what is accepted, what 145 і 45 a
− a − 45 - Pre-decubital numbers, 1 305 .
Subject: then there are no multiples of the number
Avoidance of multiples of a number
Butt 10
It is necessary to convert the LCM of negative numbers We are looking forward to replacing numbers
numbers on the back
.
Now the algorithm calculates LCM (145, 45) = 145 · 45: GCD (145, 45) = 145 · 45: 5 = 1305, having previously calculated GCD for the Euclid algorithm.
It is clear that the LCM of numbers is 145 and
one
3. If you do not divide, then check that you do not divide by deciding the number that is greater than twice, tripled, etc.
4. So check until you find the smallest number that can be divided with other numbers.
II method
2. Write a breakdown of one of the numbers (more quickly, write down the largest number).
Since the numbers are mutually prime, then the least common multiple of these numbers will be equal to zero.
Lesson progress
I. Organizational moment
II.
Usny Rakhunok
15, 67, 38, 560, 435, 226, 1000, 539, 3255.
1. Gra “I am the most important.”
Click on the bottom if the number is a multiple of 2.
Write down if the number is divisible by 5.
Stomp your feet if the number is a multiple of 10.
Why were you splashing, squeaking and stamping your feet all at once?< х < 50.
2. Name all the simple numbers that satisfy the inequalities 20
3. What is more, additional sum of these numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9?
(Amount. Additional amount is 0, and amount is equal to 45.)
4. Name the four-digit number written after the additional numbers 1, 7, 5, 8, multiples of 2, 5, 3. (1578, 1875, 1515.)
5. Marina had a whole apple, two halves and at least quarters.
How many apples did she have?
(3.)
III.
Individual work
(Dates of assignment to the studies, which were granted pardons from the independent work, allowing them to quickly complete notes in the class.)
1 card
a) 20 and 30;
b) 8 and 9;
c) 24 and 36.
2. Write down two numbers, for which the largest number will be the number: a) 5;
b) 8.
a) 22 and 33;
b) 24 and 30;
c) 45 and 9;
d) 15 and 35.
2 card
1. Find all the numbers that match the numbers and add their largest number:
a) 30 and 40;
b) 6 and 15;
c) 28 and 42.
Name a pair of mutually prime numbers like є.
2. Write down two numbers, for which the largest number will be the number: a) 3; 6 , 8, 10, 12 , 14, 16, 18 , 20, 22, 24 .
b) 9. 6 , 9, 12 , 15, 18 , 21, 24 .
3. Find the largest sibling of these numbers:
Name the smallest multiple of 2 and 3. (The smallest multiple is the number 6.)
Well, 6 years after the start of work, two boats stopped at the same time at the first pier.
How many flights can you make a skin boat in this hour?
(1 – 3 flights, 2 – 2 flights.)
How many times can I get a boat at the first pier?
(4 times)
When will you get ready?
(About the 14th year, 20th year, about the 2nd year of the night, about the 8th morning.)
IV.
Viznachennya.
The smallest natural number that is visible on the skin is called the smallest multiple.
Designation: NOC (2; 3) = 6.
The smallest divisible number that can be found is a non-prescribing divisible number.
1. Break down all the numbers into simple factors.
2. Write a breakdown of one of the numbers (precisely for the largest).
3. Supplement the given distribution with these multipliers from the distribution of other numbers that did not reach the written layout.
4. Calculate the elimination of solids.
Find the smallest multiple of the numbers:
a) 75 and 60;
b) 180, 45 and 60;
c) 12 and 35.
First you need to check that you don’t divide by any more numbers.
If so, then more will be the smallest multiple of these numbers.
Then calculate that the given numbers are mutually simple.
If so, then the smallest multiple of these numbers will be.
a) 75 is not divisible by 60 and the numbers 75 and 60 are not mutually prime, so
It’s better to write down not the number 75, but the number itself.
b) The number 180 is divisible by 45 and 60, then
LCM (180; 45; 60) = 180.
c) These numbers are mutually prime, so LCM (12; 35) = 420.
VI.
Fizkultkhvilinka
VII.
Work on the plant
1. - Write down a short note.
(There were 160 kg of apples in the warehouse in three boxes. The first box had 15 kg less, the other box had 15 kg less, the second box had 2 times more, the third box had less. How many kg of apples were there in each box?)
Solve the problem using algebra.
(Daughter’s is wearing zoshits.)
What is taken for x?
Why?
(How many kg of apples are in the third box. For x better take less.)
Todi, what can you say about box 2?
(2x (kg) apples in box II.)
Name the nutrition task.
(How many kg of apples are there in the skin box?)
Some of us wrote reports without explanation before the action, so let’s write it down briefly.
(Version: 55 kg, 70 kg, 35 kg)
2 card
2. No. 184 side. 30 (white doshki and in zoshits).
45 = 3 What do you need to earn to pay for food? 5
(Find the LCM of the numbers 45 and 60.) 5 · 3 · 3
60 = 2 ·
· 2 ·
NOC (45; 60) = 60 · 3 = 180, then 180 m.
(Vidpovid: 180 m)
VIII.
Securing the screwed material
1. No. 179 side. 30 (white doshki and in zoshits).
Find the breakdown of the simple multipliers of the smallest multiple of the numbers and the largest multiple of the numbers a and b.
a) LCM (a; c) = 3 5 7
GCD(a;c) = 5.
b) LCM (a; c) = 2 2 3 3 5 7
GCD (a; c) = 2 2 3.
2. No. 180 (a, b) side. 30 (with report comments).
a) LCM (a; b) = 2 3 3 3 5 2 5 = 2700.
b) The fragments of b are divided into a, then the LOC will be the number b itself.
LCM (a; b) = 2 3 3 5 7 7 = 4410.
(3,8 + 4,2 + 3,5 + 4,1) : 4 = 3,9
IX.
Repetition of embroidered material
1. - How to know the arithmetic mean of many numbers?
(Find the sum of these numbers; divide the result by the number of numbers.)
No. 198 side. 32 (on doshtsі and u zoshitah).
2. No. 195 side. 32 (independent).
How else can you write two numbers separately?
Inclusion in the system of knowledge and repetition
Scientists, who have begun to pay attention to an independent robot, at this stage, will conclude No. 4 RT (worker sewing, page 29) to restart the smallest multiple.
(You look like a shot.)
X. Independent work
Record intermediate lines.
Option I. No. 125 (1-2 rows) side. 22, No. 222 (a-c) side. 36, No. 186 (a, b) side. 31.
