The product of all the prime numbers less than 20 is closest to which of the following powers of 10? GMAT Explanation, Video Solution, and More Practice!

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

I really like this question. The type of thinking required for success here is exactly what you need to succeed overall on the GMAT and what we try to reinforce each and every tutoring session.

Prime numbers less than 20

Let's list out those numbers. Take your time. Double check. This isn't a race. Or at least it isn't a sprint. You'll end up saving time by doing work in a careful and considered way.

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

Just a quick note on primes. 2 is the only even prime and the smallest prime. Important thing to have memorize for the GMAT.

Ok, so we're looking to match a power of 10 with the product of these numbers. A power of 10 is 10*10*10... some number of times. So let's figure out a way to translate the product of the those primes to powers of 10. Start easy. You don't have to figure out everything all at once.

2*5 = 10. Great, we've got one 10.

3*7 = 21 which is basically 2*10

11 is basically 10

13 is a bit of a stretch but let's call it 10 for now and judge whether it's OK once we've got the first round done.

17 let's call 20 and maybe it balances out 13.

19 let's call 20 and it balances out 11.

So we've got 10*10*10*10*10*10*2*2*2*2. So 10^6*8. That's closest to 10^7.

Now, I get that you might be skeptical of the approximation BUT we did a reasonable approximation and 10^6*8 is way closer to 10^7 than 10^6. It's not close. In GMAT land I think that's good enough and I'd pick C and move on.

There is an underlying principle that proves this without a doubt that's worth knowing.

Consider:

2*100 = 200

2*101 = 202

3*100 = 300

When we added 1 to 100 we moved from 200 to 202.

When we added 1 to 2 we moved from 200 to 300.

So adding a fixed value to smaller number creates a bigger change. That makes sense since the fixed value increases the smaller by a greater percent than it increases the bigger number.

Back to our question!

2*5 = 10

3*7

11 (-1) vs 19 (+1) Given the above idea which move has more influence? The smaller one, 11.

13 (-3) vs 17 (+3) Same thing here. 13 has more influence.

So with those two we 100% underestimated. And then with 7*3 we also went down to 20. So it's clear that our approximation was even smaller than the actual number guaranteeing that the product of all the primes less than 20 is closest to 10^7.

Correct Answer: C

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

Just a quick note beyond this question. So we showed above that when adding a fixed value to a number the magnitude of the number you're adding to changes the affect of the number you add (adding a fixed value to 2 will have more impact on 2 than on a larger number say 100000).

Multiplication is different. Multiplying something by 2 doubles it's value regardless of the original magnitude (2*2 is 4, 2*100 is 200). Of course you know that. But it's important on the GMAT to put this all together in a meaningful way. We call it "absolute vs relative" and it comes up a decent amount especially on the GMAT Data Sufficiency. It tends to be that relative info (multiplication) which stays constant regardless of the magnitude of the underlying numbers is usually more helpful than absolute info, addition/subtraction.

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

Additional GMAT Puzzle Question Practice

Not the same but requires similar GMAT-puzzle-thinking. Here's a GMAT Question of the Day puzzle dealing with exponents and primes

Another similar but different puzzle question. This GMAT Question of the Day Exponents Puzzle deals more with constraints but again still tests the same type of organization skills that lead to success in the above primes question.

 

GMAT Question of the Day - Data Sufficiency - Number Properties

If x is a positive integer greater than 1 does x have more than two factors?

(1) x has the same number of factors as y^2

(2) The result when y is divided by x is an integer

AnswerShow

GMAT Question of the Day Solution

GMAT Question of the Day - Problem Solving - Puzzle

Which of the following is the greatest prime factor of 74^2 - 47^2?

A. 2

B. 3

C. 5

D. 11

E. 19

AnswerShow

GMAT Question of the Day Solution:

This is an exponent puzzle. The arithmetic would be possible but time consuming. It is rare that you would have to do so much arithmetic on a GMAT question so before you start chugging away take a pause and consider what other tools you have. With exponents there are a few things that you should consider:

1. Can you make the bases the same?

2. Can you factor?

3. Is it a special quadratic? (usually difference of squares)

In this case we have the difference of squares. You are probably used to seeing difference of squares in the format x^2 - y^2 but realize that there multiple ways that they can come up:

1. You could have a power other than two. Any even exponent will work so x^8 - y^12 is a difference of squares.

2. Real numbers also work so 74^2 - 47^2 is also a difference of squares.

3. 1 is a perfect square so 1 - y^8 would also be a difference of squares.

Get in the habit of converting difference of squares. This should be automatic. In this case, once you do that you can simplify each parenthesis so that you are only left with multiplication. Here's another place to use some GMAT sense: do not multiply out the parenthesis. We're looking for the greatest prime factor so guess what you should do: Prime Factorization. Whenever factors and multiples come up consider doing prime factorization. Make a factor tree pulling your primes out to the left (see below); this will keep your tree organized.

GMAT Exponents Quadratics

Additional GMAT Exponents Puzzle Difference of Squares Practice

Here's one of the toughest questions from the GMAT Official Guide. It's exponents based and you'll need difference of squares plus a little trick to change the format: (0.99999999/1.0001)−(0.99999991)/(1.0003) =

And another difficult GMAT Puzzle (a littles less challenging than the .99999999 question) with exponents and primes: The product of all the prime numbers less than 20 is closest to which of the following powers of 10?

Lastly, an exponents GMAT Question of the Day Puzzle.

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