Finding Factors of 12288

To find the total number of factors for any integer, we first determine its prime factorization.

The prime factorization of a number $N$ is expressed as $N = p_1^{a_1} \times p_2^{a_2} \times \dots \times p_k^{a_k}$, where $p_i$ are distinct prime numbers and $a_i$ are their positive integer exponents.

The total number of factors of $N$ is calculated using the formula: Number of Factors = $(a_1+1)(a_2+1)\dots(a_k+1)$.

Prime Factorization of 12288

Let's find the prime factors of 12288:

• $12288 \div 2 = 6144$
• $6144 \div 2 = 3072$
• $3072 \div 2 = 1536$
• $1536 \div 2 = 768$
• $768 \div 2 = 384$
• $384 \div 2 = 192$
• $192 \div 2 = 96$
• $96 \div 2 = 48$
• $48 \div 2 = 24$
• $24 \div 2 = 12$
• $12 \div 2 = 6$
• $6 \div 2 = 3$
• $3 \div 3 = 1$

The prime factorization of 12288 is $2^{12} \times 3^1$.

Calculating the Number of Factors

Using the prime factorization $12288 = 2^{12} \times 3^1$, we identify the exponents:

• Exponent of prime factor 2 ($a_1$) is 12.
• Exponent of prime factor 3 ($a_2$) is 1.

Apply the formula for the number of factors:

Number of Factors = $(12 + 1) \times (1 + 1)$

Number of Factors = $13 \times 2$

Number of Factors = $26$

Thus, the number 12288 has 26 factors.