Which of the following statements about the positive integer 180 are true? Select all that… — GRE General Test

Which of the following statements about the positive integer $180$ are true? Select all that apply.

The true statements are that

180

has

3

distinct prime factors and that the greatest common divisor of

180

and

75

15

The prime factorization of

180

2^2 \times 3^2 \times 5^1

, showing exactly

3

distinct prime factors (

2, 3, 5

). Additionally, comparing this with

75 = 3^1 \times 5^2

confirms that their greatest common divisor is

3^1 \times 5^1 = 15

Find the prime factorization of

180

180 = 2^2 \times 3^2 \times 5^1

Decomposing

180

into prime factors provides the foundation to evaluate statements regarding prime factors, GCD, and LCM.

Count the distinct prime factors of

180

The distinct prime bases are

2

3

, and

5

, giving

3

distinct prime factors.

Exponents represent multiplicity, so counting distinct bases yields the number of distinct prime factors.

Calculate the GCD of

180

and

75

Since

75 = 3^1 \times 5^2

\text{GCD}(180, 75) = 3^{\min(2,1)} \times 5^{\min(1,2)} = 3^1 \times 5^1 = 15

The GCD is found by taking the minimum exponent for each shared prime factor.

Verify definition of prime numbers and divisibility rules for remaining choices.

1

is not prime (smallest prime factor is

2

), and

\text{LCM}(180, 30) = 180

because

30

is a factor of

180

Eliminate false statements based on arithmetic definitions and divisibility properties.

Prime Factorization, Greatest Common Divisor (GCD), and Least Common Multiple (LCM)

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