Question of the Day

What is the number of distinct prime factors of the greatest common divisor (GCD) of $60$ and $90$ ?

The prime factorization of 60 is

2^2 \times 3 \times 5

and the prime factorization of 90 is

2 \times 3^2 \times 5

. The GCD is obtained by taking the lowest exponent for each common prime base:

\text{GCD}(60, 90) = 2^1 \times 3^1 \times 5^1 = 30

. The distinct prime factors of 30 are 2, 3, and 5, yielding a count of 3.

Find the prime factorizations of 60 and 90.

60 = 2^2 \times 3^1 \times 5^1

and

90 = 2^1 \times 3^2 \times 5^1

Expressing numbers as products of prime factors is the standard method for finding their GCD.

Determine the GCD by taking the lowest power of each common prime factor.

\text{GCD}(60, 90) = 2^{\min(2,1)} \times 3^{\min(1,2)} \times 5^{\min(1,1)} = 2^1 \times 3^1 \times 5^1 = 30

The GCD takes the common prime bases raised to their minimum respective exponents.

Identify and count the distinct prime factors of 30.

The distinct prime factors of 30 are 2, 3, and 5, giving a total of 3 prime factors.

By prime definition, 1 is excluded from the list of prime factors.

Greatest Common Divisor (GCD) and Prime Factorization

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