What is the total number of positive divisors of the integer 10800 that are perfect square… — State PSC Exam

What is the total number of positive divisors of the integer $10800$ that are perfect squares?

Cevap: 12

Cevap

The correct answer is found by prime factorizing

10800

into

2^4 \times 3^3 \times 5^2

. A factor is a perfect square only when its prime exponents are all even integers. Counting the available even exponents from 0 up to the maximum exponent for each prime gives 3 choices for base 2 (0, 2, 4), 2 choices for base 3 (0, 2), and 2 choices for base 5 (0, 2). Multiplying these independent choices gives

3 \times 2 \times 2 = 12

Adım Adım Çözüm

Find the prime factorization of

10800

10800 = 108 \times 100 = (2^2 \times 3^3) \times (2^2 \times 5^2) = 2^4 \times 3^3 \times 5^2

The prime factorization is necessary to systematically identify and count the divisors of the number based on exponent combinations.

Identify the mathematical condition for a divisor to be a perfect square.

A divisor

D = 2^a \times 3^b \times 5^c

is a perfect square if and only if the exponents

a

b

, and

c

are all even integers.

Perfect squares must have even exponents in their prime factorization so that their square roots are integers.

List the valid even exponent values for each prime base.

For base 2 (max exponent 4), valid even values are 0, 2, 4 (3 options). For base 3 (max exponent 3), valid even values are 0, 2 (2 options). For base 5 (max exponent 2), valid even values are 0, 2 (2 options).

These are the exponents that keep the divisor as a perfect square while still remaining a factor of

10800

Calculate the total number of perfect square divisors.

3 \times 2 \times 2 = 12

Applying the fundamental counting principle, we multiply the number of independent choices for each prime exponent.

Anahtar Kavram

Identifying perfect square factors using prime factorization

Tahmini Süre:1m 30s

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