Test: Factors - SAT MCQ

The smallest multiple of both 6 and 10 is their least common multiple (LCM). The prime factors of 6 are 2 and 3, and the prime factors of 10 are 2 and 5. The LCM is the product of the highest powers of all the prime factors: 2 x 3 x 5 = 30.

The prime factorization of 36 is 2² x 3². The number of factors is determined by adding 1 to each exponent in the prime factorization and then multiplying the results: (2 + 1)(2 + 1) = 3 x 3 = 9. 

The multiples of 4 are 4, 8, 12, 16, 20, 24, ... The multiples of 8 are 8, 16, 24, 32, ... The number 12 is a multiple of 4 but not a multiple of 8.

Since a is a multiple of 3, a = 3m for some integer m. Similarly, since b is a multiple of 4, b = 4n for some integer n. Therefore, ab = (3m)(4n) = 12mn. Since mn is an integer, ab is a multiple of 12.

The prime factorization of 24 is 2³ x 3, and the prime factorization of 36 is 2² x 3². The greatest common factor is the product of the lowest powers of the common prime factors: 2² x 3 = 12.

The prime factorization of 84 is  2 x 2 x 3 x 7.

Factors of 60 are the numbers that divide evenly into 60. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

The least common multiple (LCM) of a set of numbers is the smallest number that is divisible by all of the numbers in the set.
LCM of 8, 12, and 16 by Prime Factorization
Prime factorization of 8, 12, and 16 is (2 × 2 × 2) = 2³, (2 × 2 × 3) = 2² × 3¹, and (2 × 2 × 2 × 2) = 2⁴ respectively. LCM of 8, 12, and 16 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁴ × 3¹ = 48.
Hence, the LCM of 8, 12, and 16 by prime factorization is 48.

Assume two numbers x and y. We know that the greatest common factor (GCF) of x and y is 6, and the least common multiple (LCM) is 60.
We can use the relationship between GCF and LCM to find x and y. The relationship is given by the formula:
GCF(x, y) * LCM(x, y) = x * y
Substituting the given values, we get:
6 * 60 = x * y
360 = x * y
Now, we need to find two numbers whose product is 360 and whose GCF is 6. One possible pair of numbers is 12 and 30 because 12 * 30 = 360 and GCF(12, 30) = 6.
Therefore, the two numbers are 12 and 30.

The prime factors of 42 are 2, 3, and 7.
The prime factors of 56 are 2, 2, 2, and 7.
The prime factors of 70 are 2, 5, and 7.

Taking the common prime factors, we have 2 and 7.
Therefore, the greatest common factor of 42, 56, and 70 is 2 × 7 = 14.