Primes & Divisiblity

[Abdulla]

The product of all 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 answer is C and If I multiply 2*3*5*7*11*13*17*19 then I can find the answer.

Does anyone have faster approach to reach the answer?

[Eric_J]

I'm not sure if this is the fastest way, but here's how I did it:
first, I listed the primes (as you did)
2
3
5
7
11
13
17
19

then I multipled through the first 4, cause it was easy multiplication to get
210
11
13
17
19
then I re-wrote this as:
2.1 X 100
1.1 X 10
1.3 X 10
1.7 X 10
1.9 X 10

so, at this point, the product was equal to:
2.1 X 1.9 X 1.7 X 1.3 X 1.1 X 10^6
and then, because the answer only wants us to get the order of magnitude right, I knew I could approximate in my multiplication on each step (i.e., 1.9 X 2.1 ~ 4, which is close enough in an order of magnitude question)

4 X 1.7 X 1.3 X 1.1 X 10^6
7 X 1.3 X 1.1 X 10^6
9 X 1.1 X 10^6
10 X 10^6

which is 10^7 - answer C.

the 'trick' if any here was just combining powers of ten and then realizing that the multiplication only had to be approximate (because the answers are different enough).

there may be a faster way... anyone?

Eric

[Abdulla]

Thanks Eric_j, your approach is easier than multiplying all numbers.

[RonPurewal]

you can do some estimating here.

here's something that's relatively hard to put into words, but which is easy to illustrate: if you move the numbers in a product in offsetting (opposite) directions, then the product won't change by that much.

here's what i mean:
20 x 20 is 400.
if you increase one of these by 1 and decrease the other by 1 (i.e., move them in offsetting directions), then you get 19 x 21, which is 399. that's really close.
also, 18 x 22 = 396, 17 x 23 = 391, etc.

there are limits to this process; obviously, if you continue doing this until you're all the way down to, say, 5 x 35, you won't be anywhere close to 400 anymore. but you can make a few such 'moves' and still be all right.

so here's what i'd do:
* suck it up and multiply out 2 x 3 x 5 x 7 = 210
* take 11 x 19, lower 11 by one, raise 19 by one, and turn it into 10 x 20 = 200
* take 13 x 17, lower 13 by three, raise 17 by three, and turn it into 10 x 20 = 200
* 210 x 200 x 200 is approximately 8 with six zeroes after it, which is closest to 10,000,000 = 10^7.
voilà.

it is sine qua non here, by the way, that the answer choices are VERY far apart. if they were really close together - which would violate the spirit of the problem, since it's all about back-of-the-envelope estimation - then i would just multiply them out, approximating once they got to be more than about 3 digits long.