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Quickest Way to do this GMAT Diagnostic Test Question

This topic has 4 expert replies and 0 member replies

Quickest Way to do this GMAT Diagnostic Test Question

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Hi All,

Can someone explain the quickest and clearest way to solve this diagnostic test question?

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) $$^{10^5}$$

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GMAT/MBA Expert

Post
yasirnasir wrote:
Hi All,

Can someone explain the quickest and clearest way to solve this diagnostic test question?

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) $$^{10^5}$$
Since the answer choices are VERY spread out, 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 = 10⁷.

The correct answer is C.

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Post
Quote:
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 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

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Hi yasirnasir,

We're asked to determine which of the 5 answers is CLOSEST to the product of all the prime numbers less than 20. We can use a mix of Arithmetic and estimation to answer this question.

To start, it's worth noting that the Answers are NOT "close together" - each answer is 10 TIMES greater than the next, so we really just have to get 'close enough' with our math to answer this question.

The primes that are less than 20 are: 2, 3, 5, 7, 11, 13, 17 and 19

(5)(19) = 95 --> about 100 = 10^2
(2)(3)(17) = 102 --> about 100 = 10^2
(7)(13) = 91 --> about 100 = 10^2
(11) = about 10 = 10^1

(10^2)(10^2)(10^2)(10^1) = 10^7

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

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yasirnasir wrote:
Hi All,

Can someone explain the quickest and clearest way to solve this diagnostic test question?

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

_________________
Jeffrey Miller Head of GMAT Instruction

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