If n is an integer and 5^n 4,000,000, what is the least

Question

If n is an integer and 5^n > 4,000,000, what is the least possible value of n?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

omerrauf · Accepted Answer

Answer should be  D . Here is how I solved it.

Original Statement:  5^n>4,000,000 
so  5^n>4*10^6 
so  5^n>4*2^6*5^6

dividing by 5^6 on both sides leads to:

so  5^{n-6}>2^8 
so  5^{n-6}>256

Now we know that  5^3=125

so it has to be that the number is  5^4 , the  least  number that is greater than  256 
so  n-6=4 
so  n=10

Hence D .

Bunuel · Answer

Make prime factorization of  4,000,000=2^2*1,000,000=2^2*(10^6)=2^8*5^6=256*5^6 .

So, we have:  5^n>256*5^6  -->  5^4=625>256  (5^3=125<256), so  5^n  must be at least equal to  5^4*5^6=5^{10}  to be more than  256*5^6 .

Hence, the least possible value of n is 10.

Answer: D.

Hope it's clear.

mockney · Answer

Hi everyone,
This is an old thread, however I do have a question. Are we supposed to know value such as 2^9 off the top of our head?
Thanks for your help

Bunuel · Answer

Yes, it's good to know the powers of 2 till 2^10=1024. Check here for a solution: if-n-is-an-integer-and-5-n-4-000-000-what-is-the-least-127815.html#p1046805 Hope it helps.

ScottTargetTestPrep · Answer

Let’s break 4,000,000 into primes:

4,000,000 = 4 x 10^6 = 4 x 2^6 x 5^6 = 256 x 5^6.

So, we have:

5^n > 256 x 5^6

We see that n will definitely have to be greater than 6. We have to figure out what power of 5 is going to get us to a number greater than 256. We see that 5^3 = 125 isn’t big enough, but 5^4 = 625 fulfills our requirement of being greater than 256. Thus, n will be (6 + 4) = 10, because it is true that 5^10 > 256 x 5^6.

Answer: D

udaypratapsingh99 · Answer

Alternative way:

5=10/2

For 7 zeroes we need 10^7 and need to adjust 2^7 in the denominator, thus we need to add 3 more in power in order to maintain its value of more than 4,000,000.

kevincan · Answer

Can we use the fact that 2^10 is roughly 1,000 ?

If n is an integer and 5^n > 4,000,000, what is the least

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions