Our Lesson: Review of factors Confidential

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

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

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

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

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

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 1+2+5+4+6 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

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

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 1+2+5+4+6 or 18 is divisible by 9. … 10 if the number ends with 0. Confidential

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

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 738 + 4(4) or 754 is divisible 13. If this is not clear repeat the process again using the number 754: 75 + 4(4) or 91. If this is not clear repeat the process again using the number 91: 9 + 4(1) or 13. Confidential

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 1 + 0 + 8 = 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

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

Lets take another example Determine whether the number is divisible by 6 1. 785 No 2. 78642 yes 3. 297144 Yes Find the factors of 28 7, 3, 1 and 28 Confidential

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

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

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

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

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

Using the Sieve of Eratosthenes Confidential

Methods for finding Prime Factorization Factor Tree Method 460 10 * 46 2 * 5 23 * 2 460 = 2 * 2 * 5 * 23 Prime factors are 2² * 5 * 23 Confidential

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

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

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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

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

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

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

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

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

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

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 = 63 + 2 = 65 Expanded form Value Base Exponent 63 6 x 6 x 6 216 6 3 Confidential

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

Power to a power (xm)n = x mn (Y3)4 = y 3.4 = y12 (n2)5 = n 2.5 = n10 (3f2)3 = 33 . f2.3 = 27 f6 Confidential

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

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

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

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

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

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

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

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

Solved Examples [23/32]2 = 26/34 = 64/81 34/ 38 = 3-4 = 1/81 [23/32]2 = 26/34 = 64/81 34/ 38 = 3-4 = 1/81 46/ 42 = 44 = 256 18-2 = 1/182 = 1/324 Confidential

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

(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 t3 t-6s8 Confidential

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

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

(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

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