Question

Coach Link's assistant is trying to figure out how to organize the 35 boys and 56 girls who signed up for the skating club in groups. To do so, she needs to find the greatest common factor and least common multiple of 35 and 56.

a. Explain two ways she can find the GCF.
b. Explain two ways she can find the LCM.

Answer 1

The GCF of 35 and 56 can be found via prime factorization or the Euclidean algorithm, and both methods result in a GCF of 7. The LCM of 35 and 56 can be found by listing multiples or by dividing their product by the GCF; both methods yield an LCM of 280.

Step-by-step explanation:

To find the greatest common factor (GCF) of 35 and 56, one can use the following methods:

1. Prime Factorization: Break down both numbers into their prime factors. 35 = 5 x 7 and 56 = 2 x 2 x 2 x 7. The GCF is the product of the smallest power of common primes in the factorization, which in this case is just 7.
2. Euclidean Algorithm: Subtract the smaller number from the larger number until you get a remainder that divides both numbers. So, 56 - 35 = 21, which divides both 35 and 56, thus the GCF is 7.

To find the least common multiple (LCM) of 35 and 56, one can use the following methods:

1. List Multiples: List the multiples of each number until you find the smallest multiple they have in common. For 35: 35, 70, 105, 140, 175... For 56: 56, 112, 168... The LCM is the first common multiple, which is 280.
2. GCF and Product Method: The LCM of two numbers is equal to the product of the numbers divided by their GCF. Since the GCF of 35 and 56 is 7, and the product of the numbers is 35 x 56 = 1960, the LCM is 1960 / 7 = 280.