Factors

Question 1

When an integer is divisible by multiple numbers…?

Answer 1

1. Determine the LCM of the given numbers.
2. Examine multiples of the LCM to understand all possibilities.

Example: For divisibility by 3 and 5, the LCM is 15. Both even (30) and odd (15) multiples of 15 exist.

Question 2

Relationship between the GCF of two numbers and their sum

Answer 2

1.- The GCF of two numbers cannot be more than or equal to the sum of those two numbers.

Example: For a = 4 and b = 6, a + b = 10. The GCF of 4 and 6 is 2, which is less than 10

Question 3

What is the LCM?

Purpose - Usually, when do we use it?

*Bonus - How can we calculate the LCM

Answer 3

The smallest number that is a multiple of two numbers or more.

Example:

• The smallest number that both 12 and 18 can divide into evenly is 36. So, the LCM (12, 18) = 36

*Purpose: Usually, we use it with fractions when we want to find the smallest common denominator in a set of fractions.

*Bonus: Multiply both numbers and divide it by the GCF 12 * 18/6 = 216/6=36

Question 4

What is GCF?

Purpose - Usually, when do we use it?

Do 3 GPT exercises - Factors LCM & GCF

Answer 4

The largest number that can divide two or more numbers without leaving a remainder.

Example:

-The largest number that can divide both 12 & 18 evenly id 6. So, the GCF (12, 18) = 6

Purpose: Used when simplifying fractions to their lowest terms.

Example: 12/18 divided by 6/6 = 2/3

Question 5

What is the GCF of two none equal prime numbers?

Answer 5

1.- Prime numbers are only divisible by 1 and themselves
2.- Different prime numbers don’t share any divisors other than 1
3.- Therefore, the GCF of two distinct prime numbers is always 1

Question 6

Determining the GCF when two numbers have a unique shared factor

Answer 6

1.- All integers have at least two divisors: 1 and the number itself.
2.- If two numbers, X and Y, have only one shared factor it must be 1.
3.- Any additional shared factor would mean they share at least two factors: 1 and the other factor.

Example: If 7 and 15 share only one factor, their GCF is 1

4 and 10 share two GFC: 2 and 1.

Question 7

Given an integer A and its LCM with another integer B, how can you determine possible values for B?

Answer 7

1.- Prime Factorization A: Break down A into its prime factors
2.- Prime Factorize the LCM: Break Down the given LCM into its prime factors
3.-Analyze the prime factors: The LCM will have the highest power off all prime factors in either A or B.
4.- Determine B:
* For each prime factor in the LCM that’s not in A, B must have that factor.
* For prime factors common to both A and the LCM, if the power in the LCM is greater than A, B must account for the difference.

Example:

If A = 16 (which is 2^4) and the LCM is 48 (which is 2^4 * 3), b can be any number that introduces the factor 3 and has a power of 2 less than or equal to 2^4.

Question 8

Given an integer A and its GCF with another Integer B, how can you determine possible values for B?

Answer 8

1.- Prime Factorize A: Break down A into its prime factors
2.- Prime Factorize the GCF: Break down the given GCF into its prime factors
3.- Analyze the Prime Factors: The GCF will have the lowest power of all prime factors common to both A and b.
4.- Determine B:
*B must have all the prime factors present in the GCF
*For each prime factor in the GCF, B can have the same or a higher power of that factor.
*B can also have other prime factors not present in the GCF.

Example: If A = 16 (which is 2^4) and the GCF is 4 (which is 2^2), B can be any number that has 2^2 as a factor but doesn’t have a power of 2 greater than 2^4. B could also have other prime factors.