How do the ideas of divisibility and multiples relate to the study of fractions?
Show that 1,078 and 3,315 are relatively prime.
What is the shortest length of television cable that could be cut into either a whole number of 18-ft pieces or a whole number of 30-ft pieces?
When finding the factors of 841, what is the largest factor you would have to test? What theorem supports this?
Find the LCM of the numbers 24 and 32 by using:
a. the listing multiples method.
b. the prime factorization method.
The product of two numbers is 180. The LCM of the two numbers is 60. What is the GCF of the numbers?
Explain how you know
You know that a number is divisible by 6 if it is divisible by both 3 and 2. So why isn’t a number divisible by 8 if it is divisible by both 4 and 2?
What characteristic do the numbers 8, 10, 15, 26, and 33 have that the numbers 5, 9, 16, 18, and 24 don’t
have? (Hint: List the factors of the numbers.)
Give two more numbers that have this characteristic.
Do you think that the formula p = 6n + 1 where n is a whole number, will produce a prime number more than 50% of the time?
Give evidence to support your conclusion.
How would you like us to help you with this assignment?
1078=2 * 7 * 7 * 11
3315=3 * 5 * 13 * 17
This two numers haven't common divisors.
Thats why 1,078 and 3,315 are relatively prime
57 percent of 500
Divisibility and multiples are important concepts when studying fractions because they help determine if one number is a factor of another. Divisibility refers to whether one number can be evenly divided by another without leaving a remainder. For example, if a number is divisible by 3, it means that it can be divided by 3 without any remainder. Multiples, on the other hand, are numbers that can be obtained by multiplying a given number by another whole number. For example, the multiples of 4 are 4, 8, 12, 16, and so on.
To show that 1,078 and 3,315 are relatively prime, we need to find their greatest common divisor (GCD). The GCD is the largest positive integer that divides both numbers without leaving a remainder. If the GCD is 1, then the numbers are relatively prime. One way to find the GCD is by using the Euclidean algorithm. We start by dividing the larger number by the smaller number and taking the remainder. Then we divide the smaller number by the remainder, and continue this process until we reach a remainder of 0. The last non-zero remainder is the GCD.
To find the shortest length of television cable that could be cut into either a whole number of 18-ft pieces or a whole number of 30-ft pieces, we need to find the least common multiple (LCM) of 18 and 30. The LCM is the smallest multiple that is divisible by both numbers. One way to find the LCM is by listing the multiples of each number until we find a common multiple. In this case, we can start with 18 and 30 and list their multiples until we find a common multiple.
When finding the factors of 841, we need to determine the largest factor we have to test. Factors are numbers that divide another number without leaving a remainder. To find the factors of 841, we can start by dividing it by 2, then 3, then 4, and so on, until we reach approximately the square root of 841. The largest factor we have to test is the square root of 841. This is because any factor greater than the square root will have a corresponding factor that is less than the square root. The theorem that supports this approach is called the Fundamental Theorem of Arithmetic, which states that every positive integer can be uniquely factored into prime numbers.
To find the LCM of 24 and 32, we can use either the listing multiples method or the prime factorization method.
a. Listing multiples method: We can list the multiples of both numbers until we find a common multiple. For example, the multiples of 24 are 24, 48, 72, 96, and so on, and the multiples of 32 are 32, 64, 96, 128, and so on. The first common multiple we find is 96, so the LCM of 24 and 32 is 96.
b. Prime factorization method: We can find the prime factorization of both numbers and then take the highest power of each prime factor. The prime factorization of 24 is 2^3 * 3 and the prime factorization of 32 is 2^5. Taking the highest power of each prime factor, we get 2^5 * 3 = 96, which is the LCM.
To find the GCF of two numbers, we can use the fact that the product of the GCF and the LCM of two numbers is equal to the product of the two numbers. In this case, the product of the two numbers is 180 and the LCM is 60. So we can set up the equation GCF * 60 = 180 and solve for the GCF. Dividing both sides by 60, we get GCF = 180/60 = 3. Therefore, the GCF of the two numbers is 3.
A number is divisible by 6 if it is divisible by both 3 and 2. This is because 6 is the product of 3 and 2. However, a number is not necessarily divisible by 8 if it is divisible by both 4 and 2. This is because 8 is not the product of 4 and 2. Divisibility by 8 requires that the number be divisible by 2 three times, which is not guaranteed by the divisibility by 4 alone.
The characteristic that the numbers 8, 10, 15, 26, and 33 have but the numbers 5, 9, 16, 18, and 24 don't have is that the sum of their digits is divisible by 3. We can list the factors of each number:
Factors of 8: 1, 2, 4, 8 (not divisible by 3)
Factors of 10: 1, 2, 5, 10 (not divisible by 3)
Factors of 15: 1, 3, 5, 15 (divisible by 3)
Factors of 26: 1, 2, 13, 26 (not divisible by 3)
Factors of 33: 1, 3, 11, 33 (divisible by 3)
Two additional numbers that have this characteristic are 21 and 30. Their factors are:
Factors of 21: 1, 3, 7, 21 (divisible by 3)
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 (divisible by 3)
The formula p = 6n + 1, where n is a whole number, will not always produce a prime number. To determine if it produces a prime number more than 50% of the time, we need to analyze the values of p for different values of n. One way to do this is by testing a range of values for n, plugging them into the formula, and checking if the resulting p values are prime. Counting the number of prime p values and comparing it to the total number of values for n can give us an idea of the probability of producing a prime number. Additional statistical analysis can also be done to provide more evidence to support or refute the claim.