Finding the Smallest Integer Non-Factor

The problem asks us to identify the smallest positive integer that is not a factor of the product $P = 264 \times 90 \times 1680$. To solve this, we first need to find the prime factorization of the product P.

Prime Factorization of the Numbers

Let's find the prime factors of each number in the product:

• 264: $264 = 2 \times 132 = 2 \times 2 \times 66 = 2 \times 2 \times 2 \times 33 = 2^3 \times 3 \times 11$
• 90: $90 = 9 \times 10 = (3 \times 3) \times (2 \times 5) = 2 \times 3^2 \times 5$
• 1680: $1680 = 10 \times 168 = (2 \times 5) \times (8 \times 21) = (2 \times 5) \times (2^3 \times 3 \times 7) = 2^4 \times 3 \times 5 \times 7$

Calculating the Prime Factorization of the Product (P)

Now, we multiply these factorizations together to get the prime factorization of P:

$P = (2^3 \times 3^1 \times 11^1) \times (2^1 \times 3^2 \times 5^1) \times (2^4 \times 3^1 \times 5^1 \times 7^1)$

Combine the powers for each prime base:

$P = 2^{(3+1+4)} \times 3^{(1+2+1)} \times 5^{(1+1)} \times 7^1 \times 11^1$

$P = 2^8 \times 3^4 \times 5^2 \times 7^1 \times 11^1$

So, the prime factors of the product P are 2, 3, 5, 7, and 11.

Checking the Options

We need to find the smallest positive integer that is *not* a factor of P. Let's examine the prime factorization of the given options:

• Option 1: 12

  Prime factorization: $12 = 4 \times 3 = 2^2 \times 3^1$. Since P contains $2^8$ and $3^4$, both $2^2$ and $3^1$ are available. Thus, 12 is a factor of P.

• Option 2: 13

  13 is a prime number. Looking at the prime factorization of P ($2^8 \times 3^4 \times 5^2 \times 7^1 \times 11^1$), the prime factor 13 is not present. Thus, 13 is not a factor of P.

• Option 3: 14

  Prime factorization: $14 = 2 \times 7$. Since P contains $2^8$ and $7^1$, both $2^1$ and $7^1$ are available. Thus, 14 is a factor of P.

• Option 4: 15

  Prime factorization: $15 = 3 \times 5$. Since P contains $3^4$ and $5^2$, both $3^1$ and $5^1$ are available. Thus, 15 is a factor of P.

Determining the Smallest Non-Factor

We are looking for the smallest positive integer that is not a factor. The options given are 12, 13, 14, and 15.

• We found that 12, 14, and 15 are factors of P.
• We found that 13 is not a factor of P.

To confirm that 13 is the *smallest* such positive integer, we can consider integers less than 13:

• 1 is always a factor.
• Integers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 are all composed of prime factors (2, 3, 5, 7, 11) that are present in the prime factorization of P ($2^8 \times 3^4 \times 5^2 \times 7^1 \times 11^1$). Therefore, all these integers are factors of P.

Since all positive integers smaller than 13 are factors of P, and 13 is not a factor, 13 is the smallest positive integer that is not a factor of the product $264 \times 90 \times 1680$.