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The product of all the prime numbers less than \(\mathrm{20}\) is closest to which of the following powers of \(\mathrm{10}\)?
Let's break down what this problem is asking us to do in simple terms. We need to find all the prime numbers that are smaller than 20, multiply them all together, and then figure out which power of 10 that big number is closest to.
Remember, a prime number is a number greater than 1 that can only be divided evenly by 1 and itself. For example, 7 is prime because the only numbers that divide into it evenly are 1 and 7.
Since our answer choices are powers of 10 that are quite spread out (\(10^5\), \(10^6\), \(10^7\), etc.), we don't need to calculate the exact product - we just need a good approximation.
Process Skill: TRANSLATE - Converting the problem language into clear mathematical steps
Let's go through the numbers from 2 to 19 and identify which ones are prime:
So our prime numbers less than 20 are: 2, 3, 5, 7, 11, 13, 17, 19
Now we need to multiply: \(2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19\)
Instead of multiplying these in order, let's group them strategically to make estimation easier. We'll look for combinations that give us numbers close to powers of 10:
So our product is approximately: \(10 \times 80 \times 220 \times 60 \times 3\)
Process Skill: SIMPLIFY - Breaking down the calculation into manageable, estimatable pieces
Now let's work with our approximation: \(10 \times 80 \times 220 \times 60 \times 3\)
Let's rewrite this in terms of powers of 10:
So we have: \(10^1 \times (8 \times 10^1) \times (2.2 \times 10^2) \times (6 \times 10^1) \times 3\)
Combining the powers of 10: \(10^1 \times 10^1 \times 10^2 \times 10^1 = 10^5\)
Combining the other numbers: \(8 \times 2.2 \times 6 \times 3 = 316.8 \approx 320\)
Since \(320 \approx 3.2 \times 10^2\), our final approximation is:
\(3.2 \times 10^2 \times 10^5 = 3.2 \times 10^7\)
Since 3.2 is closer to 1 than to 10, this is closest to \(10^7\).
The product of all prime numbers less than 20 is approximately \(3.2 \times 10^7\), which is closest to \(10^7\).
The answer is (C) \(10^7\).
1. Misunderstanding "less than 20": Students may accidentally include 20 as a prime number or forget that "less than" means we stop at 19, not 20. Since 20 is not prime anyway, this particular error wouldn't affect the final answer, but it shows conceptual confusion about inequality terminology.
2. Incomplete identification of prime numbers: Students may miss some prime numbers (especially larger ones like 17 or 19) or incorrectly include composite numbers as primes. For example, they might think 9 is prime because it "looks" prime, or forget to check if 15 is divisible by 3 and 5. Missing even one prime number will significantly affect the final magnitude estimate.
3. Attempting exact calculation instead of estimation: Students may try to calculate the exact product (\(2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 19 = 9,699,690\)) rather than recognizing that the answer choices are spread far apart and only require order-of-magnitude estimation. This wastes precious time and increases chances of arithmetic errors.
1. Poor grouping choices for estimation: Students may multiply the primes in sequential order (\(2 \times 3 = 6\), then \(6 \times 5 = 30\), etc.) rather than grouping strategically. This makes mental estimation much harder and more error-prone compared to the solution's approach of grouping \(2 \times 5 = 10\) first to create a power of 10 base.
2. Errors in approximation and rounding: When estimating products like \(7 \times 11 = 77\), students may round too aggressively (calling it 100 instead of 80) or not aggressively enough. Similarly, when calculating \(13 \times 17 = 221\), they might approximate this as 200 instead of 220, leading to an incorrect final order of magnitude.
3. Mistakes in combining powers of 10: Students may incorrectly add exponents when they should multiply the bases, or vice versa. For example, when combining \(10^1 \times 10^1 \times 10^2 \times 10^1\), they might get \(10^4\) instead of \(10^5\), or incorrectly handle the coefficient multiplication (\(8 \times 2.2 \times 6 \times 3\)).
1. Misinterpreting the final coefficient: After getting \(3.2 \times 10^7\), students may incorrectly reason that since 3.2 is "much bigger than 1," they should round up to \(10^8\) instead of recognizing that 3.2 is closer to 1 than to 10, so it stays as \(10^7\). This reflects confusion about when to round up or down in scientific notation.