Chapter 6 - Extra Divisibility & Primes

Find the GCF and LCM of 100, 140, and 250.

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If two numbers have no primes in common, their GCF is

1 Source: Chapter 6- Extra Divisibility & Primes

To find the GCF and LCM of two numbers using a Venn diagram, perform the following steps:

1. Factor the numbers into primes. For example, 30 = 2 * 3 * 5 and 24 = 2 * 2 * 2 * 3 2. Create a Venn diagram 3. Place each shared factor into the shared are of the diagram (the shaded region to the right) In this example, 30 and 24 share one 2 and one 4. Place the remaining (non-shared factors in to the non-shaded areas. 5. The GCF is the product of the primes in the shared region: 2 * 3 = 6 6. The LCM is the product of all primes in the diagram: 5 * 2 * 3 * 2 * 2 = 120

Compute the GCF and LCM of 12 and 40 using the Venn diagram approach.

1. The prime factors of 12 are 2 * 2 * 3 The prime factors of 40 are 5 * 2 * 2 * 2. 2. Create the Venn Diagram 5. The GFC is the product of the primes in the shared region: 2*2 = 4 6. The LCM is the product of all primes: 3 * 2 * 2 * 5 * 2 = 120 Source: Chapter 6- Extra Divisibility & Primes

All prime numbers up 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 Source: Chapter 6- Extra Divisibility & Primes

If y = 30p, and p is pier, what is the greatest common factor of y and 14p, in terms of p?

2p Set up a chart of the prime factors of 30p and 14p. Select the lowest factor of each prime number, and multiply to find the GCF of 14p and y. Factors of 30p 2*3*5*p Factors of 14p 2*7*p GCF = 2*p

Does 36 have an odd or even number of factor pairs?

36 is a perfect square, and therefore it has an odd number of total factors.

N! must be divisible by

All integers from 1 to N

Any number that is not a perfect square will have a blank number of factors

An even number of factors

How many prime numbers exist?

An infinite number Source: Chapter 6- Extra Divisibility & Primes

All perfect squares have blank odd numbers of total factors.

An odd number of total factors.

What is the least common multiple of 18 and 24?

Answer : 72 Question type: LCM To find the LCM of two numbers, create a grid and select the highest power of each prime number. Prime factorization of 18: 2 * 3^2 24: 2^3 * 3 LCM: 72: 2^3 *3^2 = 8 * 9 Source: Chapter 6- Extra Divisibility & Primes Problem Set Q3

If a number has prime factorization of a^x * b^y * c^z, how many total factors does it have?

Answer: (x + 1)(y + 1)(z + 1)

How many total factors does 2000 have?

Answer: 20 total factors Question Type: Counting Total Factors Step 1 - Find the prime factors of 2000. 2000 = 2^4 * 5^3 Step 2 - Count the total number of possibilities for each prime number. 2^4 has 5 possibilities 2^0,2^1,2^2,2^3,2^4,2^5 5^3 has 4 possibilities 5^0, 5^1, 5^2, 5^3 Generally, a prime number has N + 1 possibilities, where is in the power of the prime. Now, multiply the total possibilities of all prime factors, which is in this case we have 3 and 5. 5 * 4 = 20 total factors

How many different prime factors does 1400 have?

Answer: 3 Question Type: Counting Factors and Primes The prime factorization of 1400 is 2^3 * 5^2 * 7 Therefore 1400 has 3 unique prime factors. Note that this could also be phrased as "unique prime factors", or "distinct prime factors"

If x and y are positive integers and x/y has a remainder of 5, what is the smallest possible value of xy?

Answer: 30 The remainder must always be smaller than the divisor. In this problem 5 must be smaller than y. Additionally, y must be an integer, so y must be at least 6. If y is 6, then the smallest possible value of xy is 30. Source: Chapter 6- Extra Divisibility & Primes Q9

How many total prime factors (length) does 252 have?

Answer: 5 Length is defined as the number of primes (not necessarily distinct) whose product is x. The prime factorization of 252 is 2^2 * 3^2 * 7 To add the total prime factors, add the exponents of the prime factors, which in this case is 5.

If the LCM of a and 12 is 36, what are the possible values of a?

Answer: 9, 18, 36 Factors of 12: 2^2, 3 Factors of 36: 3^2,2^2 Solve by creating a grid, and selecting the values that make 36 the LCM of a and 12. Source: Chapter 6- Extra Divisibility & Primes

Is the integer z divisible by 6? 1) The greatest common factor of z and 12 is 3. 2) The greatest common factor of z and 15 is 15.

Answer: A 1) The greatest common factor of z and 12 is 3. We know that 12 contains two 2s, and therefore is also divisible by 6. Since the GCF of these numbers is 3, we know that z is not divisible by 6. (Sufficient) 2) The greatest common factor of z and 15 is 15. We know that z and 15 share the greatest common factor of 15, so we know that z is divisible by 15, by 3 and by 5. We still don't know if z is divisible by 2. (insufficient) Source: Chapter 6- Extra Divisibility & Primes

If x is a prime number, what is the value of x? 1) There are a total of 50 prime numbers between 2 and x, inclusive.

Answer: A Question Type: Prime Numbers Data Sufficiency - Value Question 1) There are a total of 50 prime numbers between 2 and x, inclusive. This tells us what x must be, although we don't need to figure it out. (sufficient) 2) The is no integer n such that x is divisible by n and 1 < n < x. This is just telling us we have a prime number, which we already know. (insufficient) Source: Chapter 6- Extra Divisibility & Primes

Which of the following numbers is NOT prime? (Hint: avoid actually computing these numbers.) A) 6! - 1 B) 6! + 23 C) 6! + 41 D) 7! - 1 E) 7! + 11

Answer: B A) 6! is not prime and 1 is not prime, but 6! - 1 could be prime because they don't share any common factors. B) 6! + 21 is not prime, because 6! and 21 share a common factor of 3 because if two numbers are a factor, their sum or difference share the same factor. C) 6! + 41 do not share any factors D) 7! and 1 don't share any prime factors E) 7! and 11 don't share any prime factors

When positive integer A is divided by positive integer B, the result is 4.35. Which of the following could be the remainder when A is divided by B? A) 13 B) 14 C) 15 D) 16 E) 17

Answer: B A/B = 4 + 35/100 We know that 4 is the quotient and 0.35 is the remainder If you let R equal the remainder, then you can set up the following relationship: 0.35 = Remainder/Divisor = R / B 35/100 = R/B = 7/20 7B = 20R IN order for this equation to involve only integers, the prime factors on the left side of the equation must equal the prime factors on the right side of the equation. Therefore the remainder must be a multiple of 7! If you look at the answers, the only multiple of 7 is 14, and therefore the answer is B.

What is the value of integer x? 1) x has exactly 2 factors 2) When x is divided by 2, the remainder is 0

Answer: C 1) x has exactly 2 factors, therefore x is a prime number. Although, we don't know which prime number (insufficient) 2) When x is divided by 2, the remainder is 0. Therefore, the number must be even, otherwise the remainder would be greater than 0. We still don't know what the number is (insufficient) 3) If the number is both a prime number, and it is divisible by 2, then the number must be 2. (sufficient) Source: Chapter 6- Extra Divisibility & Primes

Is N divisible by 7? 1) N = x-y, where x and y are integers.

Answer: C Question Type: Divisibility and Addition/Subtraction 1) N = x-y, where x and y are integers. We don't know what x and y are, therefore this is insufficient. 2) x is divisible by 7, and y is not divisible by 7. We don't know how x,y, and N relate. Therefore, this is insufficient. 3) If N = x - y and x is divisible by 7 but y is not divisible by 7, then N is not divisible by 7. (Sufficient) Source: Chapter 6- Extra Divisibility & Primes

Is pq divisible by 168? 1) p is divisible by 14 2) q is divisible by 12

Answer: C Question Type: Divisibility and primes 1) p is divisible by 14. Prime factorization of 14 = 2 * 7 We don't know if pq is has a 3 (insufficient) 2) q is divisible by 12. Prime factorization of 2^2 * 3 - we don't know if pq has a 7 (insufficient) C) 14 * 12 = 168 (sufficient) Source: Chapter 6- Extra Divisibility & Primes Problem Set Q5

What is the value of integer x? 1) The least common multiple of x and 45 is 225 2) The LCM of x and 20 is 300

Answer: C Question Type: 1) x could be 25, 75, 225 (insufficient) 2) x could be 75, 150, 300 (insufficient) C) x could be 75 (sufficient Source: Chapter 6- Extra Divisibility & Primes Q7

If k^3 is divisible by 240, what is the least possible value of integer k? A) 12 B) 30 C) 60 D) 90 E) 120

Answer: C The prime box of k^3 contains the prime factors of 240, which are 2^4 * 3 * 5. Distribute the prime factors of k^3 into columns to represent the prime factors of k. Take a look at each K to see what the value of k is. 2^2 * 3 * 5 = 4 * 3 * 5 = 60

A perfect square is

Any number whose square root is two integers.

When positive integer n is divided by 7, there is a remainder of 2.

To answer this type of question, you will need to be able to list different possible values of n. Set up the integer remainder relationship Dividend = Quotient * Divisor + Remainder n = (integer) * 7 + 2 n = 2, 9, 16, 23 Notice also that the possible values of n follow a pattern: the successive value of n is 7 more than the previous value.

Finding the GCF and LCM using Prime Columns

Use this technic if you have large numbers, or you have three or more numbers. 1. Calculate the prime factors of each integer. 2. Create a column for each prime factor found within any of the integers. 3. Create a row for each integer. 4. In each cell of the table, place the prime factor raised to a power. This power counts how many copies of the column's prime factor appear in the prime box of the row's integer. To calculate the GCF, take the lowest count of each prime factor found across all the integers. This counts the shared primes. To calculate the LCM, take the highest count of each prime factor found across all the integers. This counts all primes less the shared primes.

How can you express 17 divided by 5 using remainders?

1. 17 = 3 * 5 + 2 2. 17/5 = 3 + 2/5 3. 17/5 = 3 + 2/5 = 3.4

Any integer that has an odd number of total factors must be

A perfect square

If you add two non-multiples of N, the result could be either

A multiple of N or a non-multiple of N. Example: 19 + 13 = 32 (Non-multiple of 3) + (Non-multiple of 3) = (Non-multiple of 3) 19 + 14 = 33 (Non-multiple of 3) + (Non-multiple of 3) = (Multiple of 3) Note that the exception to this rule is when N = 2. Two odds always sum to an even Source: Chapter 6- Extra Divisibility & Primes

If you add or subtract a multiple of N to a non-multiple of N, the result is...

A non-multiple of N. Example: 18 - 10 = 8 (Multiple of 3) - (Non-multiple of 3) = (Non-multiple of 3) Source: Chapter 6- Extra Divisibility & Primes

Any number whose prime factorization contains only even powers of primes must be

A perfect square Example: 144 = 2^4 * 3^2 9 = 3^2 40,000 = 2^6 * 5^4

Integer remainder relationship

Dividend = Quotient * Divisor + Remainder

When positive integer x is divided by 5, the remainder is 2. When the positive integer y is divided by 4, the remainder is 1. Which of the following values CANNOT be the sum of x and y? A) 12 B) 13 C) 14 D) 16 E) 21

Dividend = Quotient * Divisor + Remainder x = (integer) * 5 + 2 x = 2, 7, 12, 17, 22, 27 y = (integer) * 4 + 1 y = 1, 5, 9, 13, 17, 21 Answer: C - 14 A) 7 + 5 = 12 B) 12 + 1 = 13 C) ANSWER D) 9 + 7 = 16 E) 12 + 9 = 21

The prime factorization of a perfect square contains

Only even powers of primes. Example: 90^2 = (2 * 3^2 * 5)(2 * 3^2*5) = 2^2*3^4*5^2

Positive integers with exactly two factors are called

Prime numbers Source: Chapter 6- Extra Divisibility & Primes

10! + 7 must be divisible by

Question Type: Divisibility and Factorials 7, because 10! is divisible by 7 and 7 is divisible by 7.

The greatest common factor (GCF)

The large divisor of two or more integers; this factor will be smaller than or equal to the starting integers. Source: Chapter 6- Extra Divisibility & Primes

If two numbers have no primes in common, their LCM is

The product of the two numbers Source: Chapter 6- Extra Divisibility & Primes

The least common multiple (LCM)

The smallest multiple of two or more integers; this multiple will be larger than or equal to the starting integers. Source: Chapter 6- Extra Divisibility & Primes

Positive integers with more than two factors are

Never prime numbers Source: Chapter 6- Extra Divisibility & Primes

If a number's prime factorization contains any odd powers of primes, the the number is

Not a perfect square Example: 132,300 = 2^2 * 3^3 * 5^2 * 7^2