Consider the positive integer 2160. How many of its positive factors are perfect squares? — State PSC Exam

Consider the positive integer $2160$ . How many of its positive factors are perfect squares?

Cevap: 6

The prime factorization of

2160

2^4 \times 3^3 \times 5^1

. For a factor to be a perfect square, all the exponents in its prime factorization must be even integers. Analyzing the bases: the base

2

can have an exponent of

0, 2, \text{ or } 4

(three options). The base

3

can have an exponent of

0 \text{ or } 2

(two options). The base

5

can only have an exponent of

0

to remain even (one option). Multiplying the number of possibilities for each base gives

3 \times 2 \times 1 = 6

total perfect square factors.

Find the prime factorization of

2160

2160 = 2^4 \times 3^3 \times 5^1

Prime factorization is necessary to analyze the properties of the number's factors.

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

Any perfect square factor must have the form

2^{2x} \times 3^{2y} \times 5^{2z}

where the exponents are even.

Perfect squares require all prime factors to appear in pairs.

Count the number of valid even exponents for each prime factor.

For

2

: exponents

0, 2, 4

(

3

choices). For

3

: exponents

0, 2

(

2

choices). For

5

: exponent

0

(

1

choice).

The exponent of each prime in the factor cannot exceed its exponent in the original number.

Calculate the total number of perfect square factors.

3 \times 2 \times 1 = 6

The Fundamental Principle of Counting states we must multiply the independent choices to get the total number of combinations.

Using prime factorization to determine the properties and quantity of specific types of factors (perfect squares).

Soru