Understanding How to Find the Number of Ways to Express a Number as a Product of Two Factors

To find the number of ways a number can be expressed as a product of two factors, we first need to determine the total number of factors (divisors) of that number. Once we know the total number of factors, we can find the number of pairs of factors whose product equals the original number.

Prime Factorization of 8064

The first step is to find the prime factorization of 8064. We break down 8064 into its prime components:

• \(8064 = 2 \times 4032\)
• \(4032 = 2 \times 2016\)
• \(2016 = 2 \times 1008\)
• \(1008 = 2 \times 504\)
• \(504 = 2 \times 252\)
• \(252 = 2 \times 126\)
• \(126 = 2 \times 63\)
• \(63 = 3 \times 21\)
• \(21 = 3 \times 7\)
• \(7 = 7 \times 1\)

So, the prime factorization of 8064 is \(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7\).

In exponential form, this is \(8064 = 2^7 \times 3^2 \times 7^1\).

Calculating the Total Number of Factors of 8064

If a number \(N\) has a prime factorization given by \(N = p_1^{a_1} \times p_2^{a_2} \times \dots \times p_k^{a_k}\), the total number of factors (denoted by \(\tau(N)\)) is given by the product of one more than each exponent:

\(\tau(N) = (a_1+1)(a_2+1)\dots(a_k+1)\)

For \(8064 = 2^7 \times 3^2 \times 7^1\), the exponents are 7, 2, and 1.

The total number of factors of 8064 is:

\(\tau(8064) = (7+1)(2+1)(1+1) = 8 \times 3 \times 2 = 48\).

So, 8064 has a total of 48 factors.

Finding the Number of Ways to Resolve 8064 as a Product of Two Factors

Every factor \(f\) of a number \(N\) has a corresponding factor pair \(N/f\) such that \(f \times (N/f) = N\). If we list all factors, we can pair them up.

If the total number of factors \(\tau(N)\) is even, every factor \(f\) is paired with a distinct factor \(N/f\) (except when \(f = \sqrt{N}\), which only happens if \(N\) is a perfect square). In this case, the number of pairs is simply \(\tau(N)/2\).

If the total number of factors \(\tau(N)\) is odd, this means \(N\) is a perfect square, and one of its factors is \(\sqrt{N}\), which is paired with itself. When counting pairs, \(\sqrt{N}\) is counted once as part of a pair \((\sqrt{N}, \sqrt{N})\). The number of pairs is \((\tau(N)+1)/2\).

Is 8064 a Perfect Square?

A number is a perfect square if and only if all the exponents in its prime factorization are even. The prime factorization of 8064 is \(2^7 \times 3^2 \times 7^1\). The exponents are 7, 2, and 1. Since the exponent 7 and 1 are odd, 8064 is not a perfect square.

Number of Ways

Since 8064 is not a perfect square, the total number of factors (\(\tau(8064) = 48\)) is even. The number of ways to express 8064 as a product of two factors is half the total number of factors.

Number of ways = \(\frac{\tau(8064)}{2} = \frac{48}{2} = 24\).

Therefore, there are 24 ways to resolve 8064 as the product of two factors.

Revision Table: Number of Ways to Resolve a Number into Two Factors

Additional Information on Factors and Factor Pairs

Factors, also known as divisors, are integers that divide a number evenly. Finding the number of ways to express a number as a product of two factors is essentially counting the number of unique pairs \((a, b)\) such that \(a \times b = N\), where \(a\) and \(b\) are factors of \(N\).

For example, let's consider the number 12. The prime factorization is \(2^2 \times 3^1\). The total number of factors is \((2+1)(1+1) = 3 \times 2 = 6\). The factors are 1, 2, 3, 4, 6, 12. 12 is not a perfect square. The number of ways to write 12 as a product of two factors is \(6/2 = 3\). The pairs are \((1, 12), (2, 6), (3, 4)\).

Now consider the number 36. The prime factorization is \(2^2 \times 3^2\). The total number of factors is \((2+1)(2+1) = 3 \times 3 = 9\). The factors are 1, 2, 3, 4, 6, 9, 12, 18, 36. 36 is a perfect square (\(6^2\)). The number of ways to write 36 as a product of two factors is \((9+1)/2 = 10/2 = 5\). The pairs are \((1, 36), (2, 18), (3, 12), (4, 9), (6, 6)\). Notice the pair \((6, 6)\) where the two factors are the same (\(\sqrt{36}\)).