1. Our Lesson:
Review of factors
Confidential

2. WARM UP (a2/a4)-3 = a 6 {(x2v2)-2.(x6s2)} / xvs = xs/v5
Is a2/ a0 = 1 / a2 ? No
Make (-3)y = -(3)y for all values of y
Multiply Left Hand Side by (-1)y+1
Confidential

3. Factors
A factor of a number is the exact divisor of that number which means that when the latter number is divided by the former number the remainder is zero
A factor can also be expressed as a product of two whole numbers, then these two whole numbers are called factors of that number.
The number is called a multiple (product) of each of its factors
Confidential

4. 5 is a factor of 20 as it divides exactly into 20.
Lets review it with an example
If we divide 20 by 5
5 is a factor of 20 as it divides exactly into 20.
20 is the Multiple of 5
20 = 1 x 20 = 2 x 10 = 4 x 5
So, the factors of 20 are 1, 2, 4, 5, 10 and 20.
Confidential

5. Properties of factors 1. Every number is a factor of itself
2. Every factor of a number is an exact divisor of that number
3. 1 is a factor of every number
A number is the largest factor of itself or
Every factor is less than or equal to the given number
5. The number of factors of a number is finite
Confidential

6. Properties of Multiples
Every multiple of a number is greater than or equal to that number
For Example: 6 is a Multiple of each of its factors 1, 2, 3 and 6
2) The number of multiples of a number is infinite
Example: Multiples of 4 are 4, 8, 12, 16, 20, …
3) Every number is a multiple of itself
Example : 1 x 14 = 14
Confidential

7. Divisibility Rules
A number is divisible by…
… 2 if the number is even.
… 3 if the sum of the digits within the number is divisible by 3.
For example, the number 12,546 is divisible by 3 because or 18 is divisible by 3.
… 4 if the number formed by the last two digits of the number is divisible by 4.
For example,67,984 is divisible by 4 because 84 is divisible by 4. .
Confidential

8. Divisibility Rules … 5 if the number ends with either a 0 or 5.
A number is divisible by …
… 5 if the number ends with either a 0 or 5.
…6 if the number passes the divisibility tests for both 2 and
3. In other words, it must be even and the sum of its digits must be divisible by 3
… 7 if the difference of the number formed by removing the
last digit and two times that last digit is divisible by 7.
For example, 3794 is divisible by 7 because 379 – 2(4) or 371 is divisible by 7. If this is not clear repeat the process again using the number 371: 37 –2(1) or 35. …
Confidential

9. Divisibility Rules A number is divisible by …
… 8 if the number formed by the last three digits of the number is divisible by 8.
For example, 2,657,128 is divisible by 8 because 128 is divisible by 8.
… 9 if the sum of the digits within the number is divisible by 9.
For example, the number 12,546 is divisible by 9 because or 18 is divisible by 9.
… 10 if the number ends with 0.
Confidential

10. Divisibility Rules
A number is divisible by …
… 11 if the difference of the sum of every other digit within the number and the sum of the remaining digits within the number is divisible by 11.
For example, the number 40,865 is divisible by 11 because (4+8+5) – (0+6) or 17 – 6 or 11 is divisible by 11.
… 12 if the number passes the divisibility tests for both 3 and 4. In other words, the sum of the digits must be divisible by 3 and the number formed from the last two digits of the number must be divisible by 4.
Confidential

11. Divisibility Rules A number is divisible by
…13 If the sum of the number formed by removing the last digit and four times that last digit is divisible by 13.
For example, 7384 is divisible by 13 because (4) or 754 is divisible 13. If this is not clear repeat the process again using the number 754: (4) or 91.
If this is not clear repeat the process again using the number 91: 9 + 4(1) or 13.
Confidential

12. By remembering these rules we can find out factors of any number
Find Factors of 108
Lets check one by one for factors…
2: yes; It’s a even number
3: yes; as = 18 is divisible by 3
4:yes; as last 2 digits 08 are divisible by 4
5: No; the ones digit is not 0 or 5
6: Yes as the number is divisible by 2 and 3
7: No it is not divisible
Confidential

13. We get the factors as 2, 3, 6, 9 and 12
Example conti…
8: No it is not divisible
9: yes as the sum of digits is 18 and it is divisible by 9
10: No; As the ones digit is not 0
12: Yes as the number is divisible by 3 and 4
13: no as the sum of the number formed by removing the last digit and four times that last digit is divisible by 13.
We get the factors as 2, 3, 6, 9 and 12
Confidential

14. Lets take another example
Determine whether the number is divisible by 6 No
yes
Yes
Find the factors of
, 3, 1 and 28
Confidential

15. Prime factorization
A prime number is a positive integer that is not the
product of two smaller positive integers.
Note that the definition of a prime number doesn't allow 1
to be a prime number: 1 only has one factor, namely 1.
Prime numbers have exactly two factors,
When a number has more than two factors it is called a composite number.
Confidential

16. Definitions The numbers 0 and 1 are neither Prime or Composite
Prime numbers- whole number greater than 1 that is only divisible by itself and 1.
Composite Numbers – whole number greater than 1 that has more than 2 factors
Prime Factorization – expressing a composite number as a product of prime numbers
The numbers 0 and 1 are neither Prime or Composite
Confidential

17. Prime numbers from 1 to 100
2, 3, 5, 7,
11, 13, 17, 19,
23, 29,
31, 37,
41, 43, 47, 53, 59,
61, 67,
71, 73, 79,
83, 89,
97,
Sieve of Eratosthenes is an easy way to find prime numbers from 1 to 100 without checking the factors of a number
Confidential

18. Sieve of Eratosthenes
Introduced by a Greek Mathematician, Eratosthenes,
the method Sieve of Eratosthenes is an easy way to find prime numbers from 1 to 100 without checking the factors of a number
Method
List all numbers from 1 to 100
Cross out 1 because it is not a prime number
Encircle 2 and cross out all multiples of 2, other than 2 itself
Confidential

19. Steps to find prime numbers by sieve of Eratosthenes
4) We find the next uncrossed number is 3. Encircle 3
and cross out all the multiples of 3, other than 3 itself
5) The next uncrossed number is 5 . Encircle 5 and cross out all multiples of 5, other than 5 itself
6) Continue this till all numbers are either encircled or crossed out
7) All encircled numbers are prime numbers. All crossed out numbers other than 1 are composite numbers
Confidential

20. Using the Sieve of Eratosthenes
Confidential

21. Methods for finding Prime Factorization
Factor Tree Method
460 *
2 * * 2
460 = 2 * 2 * 5 * 23
Prime factors are 2² * 5 * 23
Confidential

22. prime factorization The number 60 is a composite number
It can be written as the product 2 x 2 x 3 x 5
Note that 2, 3 and 5 are factors of 60 and all these factors are prime numbers. We call them prime factors.
When we express a number as a product of prime factors, we have actually factored it completely. We refer to this process as prime factorization
Confidential

23. Lets see some examples
In the following expressions is prime factorization done?
45 = 5 x 9
Prime factorization has not been done, 9 has more factors
457 = 457 x 1
Prime factorization has been done
Confidential

24. BREAK TIME
Confidential

25. CLICK HERE TO PLAY
Confidential

26. We can find the GCF by using 2 methods
Factor
Greatest Common
Greatest Common Factor of two or more numbers can be defined as the greatest number that is a factor of each number
We can find the GCF by using 2 methods
Confidential

27. Method 1 Method 2 List the factors of each number. Then identify
the common factors.
The greatest of these common factors is the
GCF
Method 2
Write the prime factorization of each number.
Then identify all common prime factors and find their
product, to get the GCF
Confidential

28. 56 = 7 x 2 x 2 x 2 42 = 7 x 3 x 2 Lets take an example
Find the GCF of 56 and 42
Method 2
56 = 7 x 2 x 2 x 2
42 = 7 x 3 x 2
The common prime factors are 2 and 7 so the GCF is 2 x 7 = 14
Confidential

29. The exponent is sometimes referred to as the power
Exponents and
Multiplication
Exponent 46
Base
The exponent is sometimes referred to as the power
Confidential

30. Yn 43 = 4 x 4 x 4 Yn If the base (Y) is positive then the value of =
Confidential

31. The value of yn depends on whether n is odd or even
If the base (Y) is negative
The value of yn depends on whether n is odd or even
(-7)1 = -7
(-7)2 = +49
(-7)3 = -343
n =1 is odd number so value of y will be negative
n =2 is even number so value of y will be positive
n =3 is odd number so value of y is negative
Confidential

32. Laws of multiplying Bases
Rule for multiplying bases
am x an = a m + n
Product to a power
(zy)n = z n x y n
Power to a Power
(am)n = a m x n
Confidential

33. 6 x 6 x 6 x 6 x 6 = 63 + 2 = 65 Rule for multiplying bases
am x an = a m + n
6 x 6 x 6 x 6 x 6 = = 65
Expanded form
Value
Base
Exponent 63
6 x 6 x 6
216 6 3
Confidential

34. Product to a power (z y )n = z n x y n (3x)2 = 32. x2 = 9x2
(-5m)2 = (-5)2 . m2 = 25 m2
(2y)3 = 23 . Y3 = 8 y3
Confidential

35. Power to a power (xm)n = x mn (Y3)4 = y 3.4 = y12 (n2)5 = n 2.5 = n10
(3f2)3 = f2.3 = 27 f6
Confidential

36. simplifying expressions
Following the rules we have learnt
lets try and simplify the expressions
(-5e2f3g)2 = (-5)2e2.2 f3.2g2
= 25e4f6g2
Confidential

37. Lets see some examples (11a3b2)4 . b3 = (11)4 a12b11
(m0.n7.p0) . (m6.n2) = m6n9
(25xy)0 = 1
Confidential

38. 1. Quotient Law cm ÷ cn = c m - n 2. Power of a quotient law
Exponents and Division
Laws of Dividing Bases
1. Quotient Law
cm ÷ cn = c m - n
2. Power of a quotient law
(z / y)n = z n / y n
Confidential

39. 4.Power to a Power (am / bm)n = amn / bmn Laws of Dividing Bases
3. Negative Exponents
X -1 = 1 / X
4.Power to a Power
(am / bm)n = amn / bmn
Confidential

40. cm ÷ cn = cm - n cm ÷ cn = 1 c Quotient Law n - m
If c is any non-zero number and m is a larger number than n, m > n, we can write
cm ÷ cn = cm - n
In symbols if c is any non-zero number, but if n > m, we get 1
cm ÷ cn =
n - m c
Confidential

41. In this we raise a quotient or fraction to a power
Power of a quotient law
Dividing with the same exponents
In this we raise a quotient or fraction to a power
Here the power (n) is same so we multiply the z/y fraction ‘n’ number of times n n z z = n y y
Confidential

42. Negative Exponent
It indicates the reciprocal of base as a fraction (not a negative number) 1 1
X -m =
X m = x m x -m
Confidential

43. 00 Power to a Power is not allowed (am / bm)n = amn / bmn
Anything to the power of zero is equal to 1
Confidential

44. Solved Examples [23/32]2 = 26/34 = 64/81 34/ 38 = 3-4 = 1/81
[23/32]2 = 26/34 = 64/81
34/ = = 1/81
46/ 42 = = 256
= 1/ = 1/324
Confidential

45. Your turn now Find if 682 is divisible by 6 and 4 No
What is the fifth multiple of 8 40
Factorize 250 completely. 2 x 53
Can this be further factorized ? Yes
120 = 2 x 2 x 3 x 10
Find the GCF of 20 and 30 5
Confidential

46. (a2bc-2)3 a-7 = b3c-6/a 32a. a-1b2 = 9b2 32 / 34 = 1/9 4-3 / 4-5 = 16
32 / 34 = 1/9
4-3 / 4-5 = 16
t-3s4 /s-4 t t-6s8
Confidential

47. John took a loan of $300. He promised to return it in equal installments of $50. After paying two installments John was unable to pay for two months due to some accidental expenditures. He was charged an interest of $20. Find out how much does he still have to pay and in how many equal installments can he do so?
He still have to pay $225 and he can do so in 5 equal installments of $45 each
Confidential

48. If three sections of grade seven in Riverdale high are going to march in the annual parade and the strength of each section is 36, 24 and 28. Find out in how many equal rows can they march in?
4 Rows each
Confidential

49. (s2.t5)-2 /s4t-2 .
Solve the expression and find what should be multiplied with the answer to get 1 as the final answer?
Ans = s8t8
Confidential

50. You had a Great Lesson Today !
Be sure to practice what you have learned today
Confidential