The product of all the prime numbers less than...

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BTGmoderatorLU: Moderator; Posts: 2554; Joined: Sun Oct 15, 2017 1:50 pmFollowed by: 6 members

by BTGmoderatorLU » Fri Mar 23, 2018 5:42 pm

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.

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Source: Beat The GMAT — Problem Solving | Fri Mar 23, 2018 5:42 pm

Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pmFollowed by: 5 members; Thanked: 6 times

by Vincen » Sat Mar 24, 2018 3:25 am

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.

Hello LUANDATO.

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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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sat Mar 24, 2018 6:26 am

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

Brent Hanneson - Creator of GMATPrepNow.com

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8151; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Fri May 24, 2019 3:05 pm

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 â‰ˆ 10^3

Now we are left with 3, 17 and 19:

3 x 17 x 19 = 51 x 19 â‰ˆ 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

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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