This question is from official guide 12th edition.
Q> The product of all the prime numbers less than 20 is closest to which powers of 10?
(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5
Ans : C
The explanation shows we have to multiply all the primes but that takes too long. Isn't there more logical and short way?
Math problem: need help
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Here's one approach.santakdalai90 wrote:This question is from official guide 12th edition.
Q> The product of all the prime numbers less than 20 is closest to which powers of 10?
(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5
Ans : C
The explanation shows we have to multiply all the primes but that takes too long. Isn't there more logical and short way?
Since the numbers are very spread apart (each answer choice is 10 times greater than the next answer choice), we can be somewhat aggressive with our estimation.
We have the product (2)(3)(5)(7)(11)(13)(17)(19)
Let's see if we can group the numbers to get some approximate powers of 10
First (2)(5)=10, so we get (2)(3)(5)(7)(11)(13)(17)(19) = (10)(3)(7)(11)(13)(17)(19)
Next, 11 is just a bit larger that 10, so let's call it 10+
This means (10)(3)(7)(11)(13)(17)(19) = (10)(3)(7)(10+)(13)(17)(19)
Next, (7)(13)=91, which is a bit less than 100, so we get (10)(3)(7)(10+)(13)(17)(19) = (10)(3)(100-)(10+)(17)(19)
Finally, 3(17)=51, and (51)(19) is very close to (51)(20), which is very close to 1000
So,(10)(3)(100-)(10+)(17)(19) = (10)(1000)(100-)(10+)= 10,000,000 (approximately)
Since 10,000,000 = 10^7, the best answer is C
Cheers,
Brent
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Since the answer choices are VERY far apart, we can BALLPARK.santakdalai90 wrote:This question is from official guide 12th edition.
Q> The product of all the prime numbers less than 20 is closest to which powers of 10?
(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5
Ans : C
The explanation shows we have to multiply all the primes but that takes too long. Isn't there more logical and short way?
The keep things balanced, for value that we round UP, we should round another DOWN.
2*3*5*7*11*13*17*19 = ?
2*3*5*7 = 210 ≈ 200
11*13 ≈ 10*15 = 150
17*19 ≈ 15*20 = 300
200*150*300 = 9,000,000
The closest answer is 10,000,000 = 10^7.
The correct answer is C.
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santakdalai90 wrote:This question is from official guide 12th edition.
Q> The product of all the prime numbers less than 20 is closest to which powers of 10?
(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5
Ans : C
The explanation shows we have to multiply all the primes but that takes too long. Isn't there more logical and short way?
We can regroup the numbers, 2, 3, 5, 7, 11, 13, 17, and 19, to multiply them.
2 * 3 * 5 * 7 * 11 * 13 * 17 * 19
= (2 * 5) * (3 * 7) * 11 * 13 * 17 * 19
= 10 * 21 * 11 * 13 * 17 * 19
Now approximating, we get 10 * 20 * 10 * 10 * 20 * 20 = 8 * 10^6, which is [spoiler]10^7[/spoiler] appr.
The correct answer is C.
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