Problem Solving

My strategy:
2x3x5x7 would be around point something 10^2.
11x13 would also be around point something 10^2
And similarly 17x19 too would be around point 10^2
So, answer would be around 10^6
But I got confused since the number before point can also multiply & be around 10^1 or 10^2.

Any trick or tips?

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

A.10^5
B.10^9
C.10^7
D.10^6
E.10^8

Since the answer choices are VERY far apart, we can BALLPARK.
For every value that we round UP, we should compensate by rounding another value DOWN.

2*3*5*7*11*13*17*19
210 * 10*15 * 15*20
200*150*300 = 9,000,000.

The closest power of 10 = 10,000,000 = 10^7.

The correct answer is C.

Another quick and dirty approximation:

2 * 19 ≈ 40
3 * 17 ≈ 50
5 * 13 ≈ 60
7 * 11 ≈ 80

From there, we've got

40 * 50 * 60 * 80 =>

4 * 5 * 6 * 8 * 10 * 10 * 10 * 10

4 * 5 = 20 and 6 * 8 ≈ 50, so we've got

20 * 50 * 10�

2 * 10 * 5 * 10 * 10�

or 10�.

More important here is the clue that rough approximation is appropriate: the answers are far apart! Since they differ by powers of 10, we can get away with fudging the math pretty abominably.

There is a trick for doing fast math when one is multiplying numbers that are close together. For example, if someone is multiplying 11 x 13 you can simplify the math by using a reference number.

In this case, we will use the reference number of 10, because it's an easy number to work with.

(10) 11 x 13 =

We can see that eleven is higher than 10, so we can write down +1 under the 11
We can see that thirteen is higher than 10, so we can write down +3 under the 13.
Then we either add the +1 to 13 to get 14
or the +3 to 11 to get 14, and multiply that number by 10, yielding 140.
Then we multiply 1 x 3 to get 3. The final answer is 143.

This method has the advantage that it provides a pretty close estimation in the first phase, and a second phase that provides complete accuracy.

Similarly the 17x19 could be done base 20.

(20) 17 x 19 =
The 17 is -3 from 20.
The 19 is -1 from 20.
So the answer will be around (17-1)20 or (19-3)20. In either case it will be around 16x20 or 320. The exact answer is 323 because we add (-1)(-3) to 320 to get the final answer.

So fast estimation should get us:

2x5 = 10
3x7 = 21
11 x 13 ≈ 140
17 x 19 ≈ 320

And go from there.