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
The OA is C.
I'm confused by this PS question. Experts, any suggestion about how can I solve it? Thanks in advance.
The product of all the prime numbers less than...
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Hello LUANDATO.LUANDATO wrote: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
The OA is C.
I'm confused by this PS question. Experts, any suggestion about how can I solve it? Thanks in advance.
This is how I'd solve it.
All the prime numbers less than 20 are: $$2,\ 3,\ 5,\ 7,\ 11,\ 13,\ 17,\ 19.$$ Now, we can see that $$2\cdot5=10.$$ $$3\cdot7=21\approx20=2\cdot10.$$ $$11\approx10.$$ $$13\cdot17=221\approx220=22\cdot10\approx20\cdot10=2\cdot10^2.$$ $$19\approx20=2\cdot10.$$ Now, if we make the product of all the numbers we will get: $$\left(10\right)\cdot\left(2\cdot10\right)\cdot\left(10\right)\cdot\left(2\cdot10^2\right)\cdot\left(2\cdot10\right)=8\cdot10^6\approx10\cdot10^6=10^7.$$ Therefore, the correct answer is the option C.
I hope it helps.
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The numbers are very spread apart (each answer choice is 10 times greater than the next answer choice). This means we can be quite AGGRESSIVE with our estimation.LUANDATO wrote: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
The OA is C.
I'm confused by this PS question. Experts, any suggestion about how can I solve it? Thanks in advance.
One approach:
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 close enough to 10, so we get: (10)(3)(7)(11)(13)(17)(19) = (10)(3)(7)(10)(13)(17)(19) [approximately]
Next, (7)(13)=91, which is pretty close to 100.
So we get (10)(3)(7)(10)(13)(17)(19) = (10)(3)(100)(10)(17)(19) [approximately]
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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2 x 3 x 5 x 7 x 11 x 13 x 17 x 19BTGmoderatorLU wrote: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
Let's group some of these numbers to get powers of 10:
2 x 5 = 10 = 10^1
7 x 11 x 13 = 1001 ≈ 10^3
Now we are left with 3, 17 and 19:
3 x 17 x 19 = 51 x 19 ≈ 50 x 20 = 1000 = 10^3
Thus, the product of all the prime numbers less than 20 is closest to 10^1 x 10^3 x 10^3 = 10^7.
Answer: C
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