Prealg (3): Factors

Break a number down into its factors, highlighting the prime numbers as they appear 192 | \ 2 96 | \ 3 32 | \ 2 16 | \ 2 8 | \ 2 4 | \ 2 2 Therefore, 294 = 2⁶ × 3

Complete a prime factorization for 192.

8 128 has the following pairs of factors, for a total of 8: 1) 1, 128 2) 2, 64 3) 4, 32 4) 8, 16

How many factors does the number 128 have?

Break a number down into its factors, highlighting the prime numbers as they appear 294 | \ 2 147 | \ 3 49 | \ 7 7 Therefore, 294 = 2 × 3 × 7 × 7, or 2 × 3 × 7²

Complete a prime factorization for 294.

Break a number down into its factors, highlighting the prime numbers as they appear 495 | \ 5 99 | \ 3 33 | \ 3 11 Therefore, 495 = 3 × 3 × 5 × 11, or 3² × 5 × 11

Complete a prime factorization for 495.

12a⁴b³c⁴ I) Start by getting the GCF of the coefficients (in this case, 24 and 36). The fastest way to find a GCF is to take the smaller number: 1) Break 24 down into its factors a) 1, 24 b) 2, 12 c) 3, 8 d) 4, 6 2) Now test whether each factor, from highest to lowest number, also goes into the larger number (in this case, 36) a) 1, 24 b) 2, 12 c) 3, 8 d) 4, 6 12 goes into 36, so it is the GCF of the coefficients. II) The GCF of variables: circle the one with the smaller exponent a⁴ vs. a⁷ b⁸ vs. b³ c⁷ vs. c⁴ 3) Putting it all together, the GCF of 24a⁴b⁸c⁷ and 36a⁷b³c⁴ is: 12a⁴b³c⁴

Find the GCF of 24a⁴b⁸c⁷ and 36a⁷b³c⁴

27yz⁵ I) Start by getting the GCF of the coefficients (in this case, 81 and 54). The fastest way to find a GCF is to take the smaller number: 1) Break 54 down into its factors a) 1, 54 b) 2, 27 c) 3, 18 d) 6, 9 2) Now test whether each factor, from highest to lowest number, also goes into the larger number (in this case, 81) a) 1, 54 b) 2, 27 c) 3, 18 d) 6, 9 27 goes into 81, so it is the GCF of the coefficients. II) The GCF of variables: circle the one with the smaller exponent x² vs. 1 (the second term doesn't have x, so we use 1) y³ vs. y z⁸ vs. z⁵ 3) Putting it all together, the GCF of 81x²y³z⁸ and 54yz⁵ is: 27yz⁵

Find the GCF of 81x²y³z⁸ and 54yz⁵

8 54 has the following pairs of factors, for a total of 8: 1) 1, 54 2) 2, 27 3) 3, 18 4) 6, 9

From The College Panda ACT Math: Advanced Guide and Workbook (p. 33)¹: How many factors does the number 54 have?

D) 20 If we translate the statement "A number z is produced by multiplying x, which is divisible by 10, and y, which is divisible by 15" ...into an equation, we would get: xy = z If x is divisible by 10 and y is divisible by 15, then z must be divisible by both 10 and 15 (or 150 all together). Therefore, we must look for the option that is NOT a factor of 150. Only 20 does not go into 150, so the answer is (D)

From The College Panda ACT Math: Advanced Guide and Workbook (p. 36)¹: A number z is produced by multiplying x, which is divisible by 10, and y, which is divisible by 15. All of the following whole numbers must be factors of z EXCEPT: A) 2 B) 3 C) 6 D) 20 E) 30

B) 552 If you translate the statement "A positive integer is multiplied by 6, and the product is multiplied by 4" ...into an equation, you would get: x * 6 * 4 = y Where x = a positive integer y = the result Simplified, the equation is: 24x = y Rearranged, we get: x = y/24 Notice how this means that y must be divisible by 24, since we get x, which is a positive integer. Therefore, we must choose the option that is divisible by 24. Only 552 works, so the answer is (B)

From The College Panda ACT Math: Advanced Guide and Workbook (p. 36)¹: A positive integer is multiplied by 6, and the product is multiplied by 4. Which of the following numbers could be the result after the second multiplication? A) 540 B) 552 C) 564 D) 588 E) 612

Break a number down into its factors, highlighting the prime numbers as they appear 45 | \ 5 9 | \ 3 3 Therefore, 45 = 3 × 3 × 5, or 3² × 5

How do you do a prime factorization? Complete one for 45.

8x³y⁷z³ I) Start by getting the GCF of the coefficients (in this case, 72 and 16). The fastest way to find a GCF is to take the smaller number: 1) Break 16 down into its factors a) 1, 16 b) 2, 8 c) 4, 4 2) Now test whether each factor, from highest to lowest number, also goes into the larger number (in this case, 72) a) 1, 16 b) 2, 8 c) 4, 4 8 goes into 72, so it is the GCF of the coefficients. II) The GCF of variables: circle the one with the smaller exponent x⁴ vs. x³ y⁹ vs. y⁷ z³ vs. z⁵ 3) Putting it all together, the GCF of 72x⁴y⁹z³ and 16x³y⁷z⁵ is: 8x³y⁷z³

How do you find the GCF of rational expressions? Use your knowledge to find the GCF of 72x⁴y⁹z³ and 16x³y⁷z⁵

8 144 has the following pairs of factors, for a total of 15: 1) 1, 144 2) 2, 72 3) 3, 48 4) 4, 36 5) 6, 24 6) 8, 18 7) 9, 16 8) 12, 12

How many factors does the number 144 have?

Any integer that "goes into" another number evenly For example: 1, 2, 3, 4, 6, and 12 are factors of 12 ...because 12 can be divided by those numbers without a remainder

What is a factor?

The largest factor that a set of numbers has in common. The fastest way to find a GCF is to take the smaller number: 1) Break 12 down into its factors: a) 1, 12 b) 2, 6 c) 3, 4 2) Now test whether each factor, from highest to lowest number, also goes into the larger number (in this case, 18) a) 1, 12 b) 2, 6 c) 3, 4 12 does not go into 18, but 6 does. Therefore, 6 is the GCF

What is a greatest common factor (GCF)? Use your knowledge to find the GCF of 18 and 12

16 The fastest way to find a GCF is to take the smaller number: 1) Break 32 down into its factors a) 1, 32 b) 2, 16 c) 4, 8 2) Now test whether each factor, from highest to lowest number, also goes into the larger number (in this case, 144) a) 1, 32 b) 2, 16 c) 4, 8 32 does not go into 144, but 16 does. Therefore, 16 is the GCF

What is the GCF of 144 and 32?

8 The fastest way to find a GCF is to take the smaller number: 1) Break 248 down into its factors a) 1, 248 b) 2, 124 c) 4, 62 d) 8, 31 2) Now test whether each factor, from highest to lowest number, also goes into the larger number (in this case, 256) a) 1, 248 b) 2, 124 c) 4, 62 d) 8, 31 None of the larger numbers go into 256, but 8 does. Therefore, 8 is the GCF

What is the GCF of 256 and 248?

MATH - NUM - gcd(56,120) The GCF is 8

What's the calculator shortcut for GCF? Use your knowledge to find the GCF of 56 and 120