Problem Solving

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.

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.

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

BTGmoderatorLU 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

2 x 3 x 5 x 7 x 11 x 13 x 17 x 19

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