NCERT Class 6 Maths Chapter 3: Playing with Numbers: Breaking the Numbers

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Questions?

1. Is 5 a multiple of 5 or its factor?

2. Explain fundamental theorem of arithmetic?

3. Can you find all the factors of a number? What about its multiples?

4. Do larger number has more factors?

Overview

  • Factors
  • Factor Tree & Properties
  • Properties of Factors
  • Prime Factors
  • Perfect Numbers
  • Fundamental Theorem of Arithmetic
  • Multiples and Properties
  • Common Factors
  • Common Multiples
  • Coprime Numbers
  • HCF & LCM & Word Problems
  • Prime and Composite Numbers
    • The Sieve of Eratosthenes
    • Special Prime Numbers
  • Divisibility Tests
    • Look at Digits
    • Sum of Digits
    • Last Digits
    • Combinations
    • Uses of Divisibility Tests

Breaking the Numbers?

Breaking Numbers
Breaking Numbers
Breaking Numbers
Breaking Numbers

Breaking the Numbers by Addition

Breaking by Addition

We can create any number just by adding right number of 1՚s

Breaking the Numbers by Multiplication

Breaking by Multiplication

We can also break numbers by multiplication- constituents are known as factors. The original number is called the multiple of its factors.

Arranging in Row Column

Arranging
  • This is represented by a simple factor tree.
  • At top we have root number we which so express as product of numbers and at the last level we have leaves.
Arranging

This is another representation of number 6 expressed as tree.

Arranging
Arranging

Factors of Number 6

Factors of 6
  • 6 can be represented as product of 2 numbers in several ways. The various numbers which constitute 6 are called the factors. The process of dividing the number into its factors is called factorization
  • At the leaf level we have factors of number 6. Factors are numbers which can evenly divide a given number.
  • 6 is a multiple of all its factors (that is it appears in the tables of all these numbers)
  • For 12 we have several factors 3, 4, 1,12, 6 etc. 12 is the multiple of all its factors.
  • At the same time multiples of 12 are include 12,24, 36 and so on … .

The Factor Tree

Factor Tree
  • The factor tree divides the root number into factors.
  • The product of all the numbers at any level gives us the root number 24.
  • There are 2 kinds of factors of number- prime and non-prime.
  • The red boxes show the prime factors of 24. The blue boxes show the non-prime factors of number 24.

Properties: Each Subtree is Complete Factor Tree

Factor Tree

Each subtree is complete factor tree in itself

Properties: Nodes Are Factors of Root

Factors of Root

All the numbers the below the number in the subtree are factors of root- not all the factors but only some.

Properties: Uniqueness, Completeness & Leaves

Uniqueness, Completeness

Here are some possible factor trees of number 24.

1. A factor tree is not unique

2. The factor tree does not show all the factors of 24.

3. For example, 6 only appears in first tree, 12 in second and 8 in third.

4. What is unique is the prime factors of number 24- the ones listed in pink boxes in the leaf nodes- the path to reach these prime factors is different.

Properties of Factor Tree: Summary

Summary
Summary

Here are some possible factor trees of number 24.

1. Product at each level gives root number

2. A factor tree is not unique

3. The factor tree does not show all the factors of 24- For example 6 only appears in first tree, 12 in second and 8 in third.

4. What is unique is the prime factors of number 24- the ones listed in pink boxes in the leaf nodes- the path to reach these prime factors is different.

Properties of Factors

Factors
Properties of Factors
Properties of Factors

Factors and Prime Factors

Factors Prime Factors
  • Find all the factors and prime factors of 84.
  • Prime factors – 2,2, 3,7 (do not include 1 and the number, unless it is prime)
  • The process of dividing the number into its prime factors is called prime factorization

All factors- Combine the prime factors in various ways

  • 1
  • 2,2, 3,7
  • 4, 6,14, 21
  • 12,42, 28
  • 84

Perfect Numbers

Perfect Numbers
  • 28 (1 + 2 + 4 + 7 + 14 + 28 = 56) is perfect
  • 14 (1 + 2 + 7 + 14 = ) is deficient
  • 12 (1 + 2 + 3 + 4 + 6 + 12 = ) is abundant

From Common Multiples to LCM, Calculating LCM & Word Problems

From Common Factors to HCF, Calculating HCF & Word Problems

Prime Factors Are Unique

Prime Factors Unique

From the factor tree we understand that although the factor tree is not unique it always lists all the prime factors of a number.

Fundamental Theorem of Arithmetic

  • Every integer greater than 1 either is a prime number itself or can be uniquely represented as the product of prime numbers
  • A prime number has only two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite – because it has only one factor (1)
  • The prime factors and their count provides a unique signature
  • 1200 = 24 × 31 × 52
  • For example, 1200 can be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product.
  • The requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique (e. g. , 24 = 8 × 3 = 12 × 2) .
  • This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique; for example, 2 = 2 × 1 = 2 × 1 × 1 = …
  • Additive structure of natural numbers is pretty simple just add 1 to get next number. The multiplicative structure is governed by primes.

Multiples

Multiples

Multiples Are Found up the Tree

Multiples
  • Any path up the tree is made up of multiples of the number.
  • We go up to parent then grandparent etc. they are all multiples of their children and grandchildren.

Properties of Multiples

Properties of Multiples
Multiples

Common Factors and HCF

Common Factors and HCF

These are only few common factors of the 84 and 42.

There are more lets list all the factors:

84: 1, 2, 2, 3, 7, 4, 6, 14, 12, 21, 42,28, 84

24: 1, 2, 2, 3, 4, 6, 8,12, 24

  • There are only a finite common factors of 84 and 42.
  • Now consider the prime factors of these 2 numbers- 2,3 and 2. Only these 3 prime factors are found in both 84 and 24 (fundamental theorem of arithmetic) . It means 2 x 3 x 2 = 12 is the largest number which can divide both the numbers 84 and 24. This is known as HCF or highest common factor.
  • HCF has word highest in it but is smaller than or equal to each of the numbers. It is the largest number which can evenly divide given numbers.

Coprime Numbers

Coprime Numbers
  • Some numbers although they are not prime have no common factors- they are known as co-primes.
  • That is their LCM is always 1.
  • Note that primes numbers are always co-primes.

Common Multiples

Common Multiples
  • When we look at multiples, they are simply the tables of numbers. let՚s take 2 - 9 and 12.
  • Common multiples are reached when 12 goes ahead 3 multiples and 9 moves ahead by 4 multiples.
  • First common multiples is 36 and thereafter all multiples of 36 are common to both 12 and 9.
  • There are endless common multiples of any number
  • The number 36 is very important it is the smallest common multiple between 12 and 9- called the least common multiple of LCM
  • LCM has word lowest in it but is bigger than or equal to each of the numbers. It is the smallest number which is divisible by all the given numbers
  • Note that LCM of prime numbers is just the product of numbers

Calculating Highest Common Factor

  • Include factors common to all the numbers once.
  • Include a factor many times- if it appears multiple times in all the numbers.

126: 2,3, 3,7; 6, 9,14, 21; 18,42, 63; 126

252: 2, 2,3, 3,7; 4, 6, 9,14, 21; 12, 18, 28,42, 63; 84,126

315: 3,3, 7,5; 9, 15,21, 35; 45,63, 105; 315

126: 2,3, 3,7

252: 2, 2,3, 3,7

315: 3,3, 7,5

Calculating Highest Common Factor
126252315
34284105
3142835

HCF = 9

HCF has word highest in it but is smaller than or equal to each of the numbers. It is the largest number which can evenly divide given numbers.

Properties of HCF

  • Is a factor of all the numbers (HCF) .
  • HCF is greatest number which can divide given numbers (GCD) .
  • Smaller than or equal to given numbers (6,12) .
  • Equal to 1 for coprime (including prime) numbers.

Word Problems on HCF

Word Problems
  • Find the largest scale using which we can measure the length of these walls.
  • Of course, scale of length 1 would work.
  • 2 would also work.
  • The idea is simple we can go as long as the HCF of the numbers because that is the largest number which would divide the lengths of both the walls evenly.
  • Bigger number such as 8 would not divide both 12 and 18.
  • The length, breadth and height of a room are 825 cm, 675 cm and 450 cm, respectively. Find the longest tape which can measure the three dimensions of the room exactly.
Word Problems on HCF
5825675450
516513590
3332718
1192

HCF = 5 ⚹ 5 ⚹ 3 = 75

Calculating Least Common Multiple

Least Common Multiple
  • Include factors common to at least two numbers only once.
  • Now include the factors from all the numbers not yet included.
  • Include factors multiple times if necessary.
  • 42: 2,3, 7
  • 105: 5,3, 7
  • 245: 5,7, 7
Calculating Least Common Multiple
42105245
5422149
7637
3217
  • We stop here LCM 5 x 7 x 3 x 2 x 7 = 1470
  • Include common factors only once- but include all the factors which appear in at least one factorization.
  • Include repeated factors twice
  • Thus, we get, 2 x 3 x 7 x 5 x 7

Properties of LCM

  • Smallest number divisible by all the numbers.
  • Greater than or equal to given numbers (6,12) .
  • Equal to product of coprime (including prime) numbers.

Word Problems on LCM

LCM
  • LCM means Lowest common multiple Consider 2 scales of length 6 and 8 cm.
  • Find the min length of wall which would allow us to measure length evenly using either scales.
  • LCM means Lowest common multiple Consider 2 scales of length 6 and 8 cm.
  • Find the min length of wall which would allow us to measure length evenly using either scales.
  • LCM = 2 ⚹ 3 ⚹ 3 = 18
  • Therefore 18 would be evenly divided by both 6 and 9 and hence would be required length of wall.
  • Determine the smallest 3-digit number which is exactly divisible by 6,8 and 12.
  • Determine the smallest 3-digit number which is exactly divisible by 6,8 and 12.
  • LCM = 6,8, 12 = 24. Multiples of 24 will be all be divisible by all three numbers, 24, 48, 72,96, 120. The smallest 3 digit number is thus 120.
  • Find the least number which when divided by 6,15 and 18 leave remainder 5 in each case.
  • Find the least number which when divided by 6,15 and 18 leave remainder 5 in each case.
  • LCM of 6,15, 18 is 90.90 is smallest number divisible by all three numbers with remainder 0. Thus 91 would leave 1 as remainder when divided by each of the numbers and 95 would give a remainder of 5.
  • 3 boys step off together from the same spot. Their steps measure 63 cm, 70 cm and 77 cm, respectively. Find minimum distance so that all can cover the distance in complete steps?
  • We need a number divisibly by 63,70 and 77
Word Problems on LCM
637077
791011

LCM = 7 ⚹ 9 ⚹ 10 ⚹ 11 = 6930

Difference in Calculating LCM & HCF

LCM vs. HCF

LCM of 12,18, 27

Difference in Calculating LCM & HCF
2121827
36927
3239
213

LCM = 2 x 3 x 3 x 2 x 3 = 108

HCF of 12,18, 27

Difference in Calculating LCM & HCF
3121827
469

HCF = 3

The Factor Tree

The Factor Tree
  • At the leaf level we have factors of number 6. Factors are numbers which can evenly divide a given number.
  • 6 is a multiple of all its factors (that is it appears in the tables of all these numbers)
  • Two of the factor trees are not so interesting- they are formed by number itself and 1 at the bottom- this can keep on going on. The other two are more interesting.

Breaking the Numbers!

Breaking Numbers

When a number can be represented in terms of the factors other than 1 and itself, we say that the number is composite.

Can We Break 3 or 5

Break 3 or 5

Prime Numbers

  • Numbers Like 3 & 5 Which Only have 1 and themselves as Factors (Exactly 2 factors)
  • 1 Is Neither Prime Nor Composite
  • A prime number has only two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite – because it has only one factor (1)
Prime Numbers

Composite Numbers

Numbers Like 6 Which Have Factors Other than 1 and themselves (More Than 2 Factors)

Composite Numbers

A composite number has more than 2 factors

The Factor Game: Prime Number Trick

Factor Game
Prime Number Trick
  • The student quickly realizes that we need to pick a prime number as large as possible- to make our total large while not benefitting opponent.
  • Another way would be to pick the largest possible number all of whose factors have already been removed

2 Relations between Primes & Composites

  • All the multiples of prime numbers are composite numbers.
  • Composite numbers can be expressed as product of primes.
Primes Composites

The Sieve of Eratosthenes

Sieve of Eratosthenes
  • He is best known for being the first person to calculate the circumference of the Earth.
  • He was also the first to calculate the tilt of the Earth՚s axis, once again with remarkable accuracy.
  • The sieve works by marking the composite numbers- filtering them out. What is left on top are the prime numbers.
  • A prime number has only two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite – because it has only one factor (1)
Sieve of Eratosthenes

A prime number has only two factors: 1 and itself. A composite number has more than two factors.

The number 1 is neither prime nor composite – because it has only one factor (1) - so first we remove 1.

Sieve of Eratosthenes
Sieve of Eratosthenes
  • Now we move on to number 2.
  • We know that any of the multiples of 2 except 2 cannot be prime- because one of their factors is 2 (in addition to 1 and itself) .
  • So, we move through multiples of 2 and mark them as composites.

Now we move to number 3

Sieve of Eratosthenes

1. If 3 was a factor of any of the numbers seen so far- it would have been cancelled. But it is not so the only factors of 3 are 3 and 1- and hence it is prime. Therefore, if we reach any number and it has not been cancelled it has to be prime. So, we mark three as prime.

2. We move through 3 and cancel all its multiples as they cannot be prime.

Sieve of Eratosthenes
  • let՚s look at number 4. It has been marked as composite because it had 2 as factor. But what about its multiples?
  • We don՚t have to consider its multiples because they would have been cancelled already as they would have been multiples of factors of 4.
  • Now we look at 5. Since it has not been cancelled, we mark it prime and then mark all its multiples as composites.
  • We only need to do this process till 10. Because if there were a multiple of number greater than 10 let՚s say 11, its other factor would have to be between 1 and 9.11 ⚹ 10 is 110 which is already more than 100.

Special Primes

Special Primes

Twin Primes

  • Difference is 2
  • (3,5) , (5,7) , (17,19)

Prime Triplets

  • 3 primes where smallest & largest differ by 6
  • (5,7, 11) , (7,11, 13)

2 is the smallest prime number which is even.

  • Every prime number except 2 is odd.
  • Twin primes: (3,5) , (5,7) , (11,13) , (17,19)
  • Usually the pair (2,3) is not considered to be a pair of twin primes. Since 2 is the only even prime, this pair is the only pair of prime numbers that differ by one; thus, twin primes are as closely spaced as possible for any other two primes. All the twin primes except 3,5 is divisible by 6.5 is a member of 2 pairs
  • Prime triplets: Prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form (p, p + 2, p + 6) or (p, p + 4, p + 6) . With the exceptions of (2,3, 5) and (3,5, 7) , this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime (except for 3 itself)
  • Ex: (5,7, 11) , (7,11, 13) , (11,13, 17)

There are also prime quintuplet and prime quadruplets.

Is a Number Prime?

Is a Number Prime?
  • Finding out if a number is prime is a hard problem- even for a computer- especially as the numbers get large. let՚s see why that is so- 199. Since sq root of 199 is approximately 14.1 we need to check only till 15.
  • What about 583? We have to check prime numbers till 24 (11 ⚹ 53) - things get especially troubling when a number is formed by multiplication of two large prime numbers- this problem is so hard even for computers that it forms the basis of computer security.
  • To help us determine if number is prime by trial and error, we can use the aid of divisibility tests.

Divisibility Tests: Look at Digits

2: Number is even

5: Number ends in 5 or 0

10: Number ends in 0

3: Sum of digits is divisible by 3

9: Sum of digits is divisible by 9

11: Sum of odd digits -Sum of even digits is 0 or divisible by 11

  • 61809 is divisible by 3 (sum is 24) ()
  • 61809 is not divisible by 9
  • 61809 + 3 = 61812 is divisible by 9 (18) ()
  • If abcd is divisible by then 1000a + 100b + 10c + d is divisible by is divisible by . Each of () is divisible by . This remainder if any would also come from
  • . Now are divisibly by 11. Therefore have to be divisible by 11 to make abcd divisible by 11.

Divisibility Tests: Last Digits

4: Last 2 digits are divisibly by 4

8: Last 3 digits are divisibly by 8

Divisibility Tests: Combinations

  • Number is divisible by 2 co-prime numbers then it is divisible by their product
  • A number is divisible by a given divisor if it is divisible by the highest power of each of its prime factors.
  • If a number is divisible by two co-prime numbers, then it is divisible by their product also.
  • For example, 18 is divisible by both 3 and 2 and therefore it is divisible by 6 as well.
  • Similarly, if a number is divisibly by both 9 and 4 then it is divisible by 36 as well. For example, 108 (36 ⚹ 3) .
  • Along the same lines a number is divisible by a given divisor if it is divisible by the highest power of each of its prime factors. For example, to determine divisibility by 36, check divisibility by both 4 and by 9 (they are coprime) . Checking divisibility by 12 and 3 is not sufficient- they are not coprime (for example 24 is divisibly by 12 and 3 but not by 36) . However, number 72 is divisible by both 4 (4 ⚹ 18) and 9 (8) and hence is divisible by 36 as well (36 ⚹ 2) .
  • If a number is divisible by another number, then it is divisible by each of the factors of that number.
  • If two given numbers are divisible by a number, then their sum & difference is also divisible by that number.
  • If a number is divisible by another number, then it is divisible by each of the factors of that number.
  • For example, 12 is divisible by 6 therefore it is also divisible by both 3 and 2.
  • If two given numbers are divisible by a number, then their sum is also divisible by that number.
  • For example, both 6 and 9 are divisible by 3. Therefore 9 + 6 = 15 is also divisible by 3 (distributive law)
  • If two given numbers are divisible by a number, then their difference is also divisible by that number.
  • For example, both 12 and 20 are divisible by 4. Therefore 20 - 12 = 8 is also divisible by 4 (distributive law)

Uses of Divisibility Tests

  • Find if a number is prime.
  • Do the prime factorization.
  • Find if number 429 is prime (divisible by 3) .
  • Find the prime factorization of 1749 = 1749/3 = 583 (5 + 3 – 8) = 0 (divisible by 11) , 583/11 = 53.
  • There are divisibility tests for other numbers as well.

More Advanced Topics

1. Remainder theorem.

2. Euclid՚s division lemma.

3. Factorization of prime numbers.

4. Advanced divisibility tests.

5. More kind of remainder word problems based on LCM/HCF.

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