10,008 : 24

To simplify the ratio \(10,008 : 24\), we can divide both numbers by their greatest common divisor (GCD).

First, we can find the GCD of 10,008 and 24.

The prime factorization of 24 is: \[ 24 = 2^3 \times 3^1 \]

Now, let's factor 10,008:

• 10,008 is even, so we divide by 2: \[ 10,008 ÷ 2 = 5,004 \]
• 5,004 is also even, so we divide by 2 again: \[ 5,004 ÷ 2 = 2,502 \]
• 2,502 is even, so we divide by 2 again: \[ 2,502 ÷ 2 = 1,251 \]

Now, 1,251 can be divided by 3: \[ 1,251 ÷ 3 = 417 \] Next, 417 can also be divided by 3: \[ 417 ÷ 3 = 139 \]

So, the factorization of 10,008 is: \[ 10,008 = 2^3 \times 3^2 \times 139^1 \]

Now, the GCD between \(10,008\) and \(24\) would be found from their common factors.

The common factors:

• For \(2\), the minimum power is \(2^3\) from both.
• For \(3\), the minimum power is \(3^1\) from \(24\).

Therefore, the GCD is: \[ GCD(10,008, 24) = 2^3 \times 3^1 = 24 \]

Now we can simplify the ratio: \[ 10,008 ÷ 24 : 24 ÷ 24 \] \[ 417 : 1 \]

Thus, the simplified ratio is: \[ 417 : 1 \]