Question

The product of all the prime numbers less than 20 is closest to which of the following powers of 10?

• A. \(10^9\)
• B. \(10^8\)
• C. \(10^7\)
• D. \(10^6\)
• E. \(10^5\)

Hint:

In estimation problems, strategically group numbers to create easy multiples (like 10 from 2 and 5). You can often get close enough to the answer with good approximations without needing to perform the full, precise calculation.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to identify all prime numbers less than 20, calculate their product, and then determine which power of 10 is the closest approximation to this product.
Step 2: Detailed Explanation:
First, list the prime numbers less than 20: \[ 2, 3, 5, 7, 11, 13, 17, 19 \] Next, calculate their product (P). We can group numbers to make the calculation easier: \[ P = (2 \times 5) \times (3 \times 7) \times 11 \times 13 \times 17 \times 19 \] \[ P = 10 \times 21 \times 11 \times 13 \times 17 \times 19 \] Now let's continue multiplying: \[ P = 210 \times 11 \times 13 \times 17 \times 19 \] \[ 210 \times 11 = 2310 \] \[ P = 2310 \times 13 \times 17 \times 19 \] \[ 2310 \times 13 = 30030 \] \[ P = 30030 \times 17 \times 19 \] \[ 30030 \times 17 = 510510 \] \[ P = 510510 \times 19 \] Let's approximate \(510510 \times 19 \approx 510000 \times 20 = 10200000\), which is \(1.02 \times 10^7\). This suggests the answer is close to \(10^7\). Let's do the exact calculation: \[ 510510 \times 19 = 510510 \times (20 - 1) = 10210200 - 510510 = 9699690 \] The product is 9,699,690.
Step 3: Comparison with Powers of 10:
We need to see if 9,699,690 is closer to \(10^6\) or \(10^7\). \[ 10^6 = 1,000,000 \] \[ 10^7 = 10,000,000 \] Distance from \(10^7\): \(|10,000,000 - 9,699,690| = 300,310\).
Distance from \(10^6\): \(|9,699,690 - 1,000,000| = 8,699,690\).
The product is much closer to 10,000,000 (\(10^7\)).
Step 4: Final Answer:
The product is closest to \(10^7\).