Number theory

[Prajakt]

If n is an integer, and 5^n > 4,000,000 then what is the least possible value of n ?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

[Anurag@Gurome]

If n is an integer, and 5^n > 4,000,000 then what is the least possible value of n ?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

Prime factorization of 4,000,000 = 2² * 1,000,000 = 2² * 10^6 = 2^8 * 5^6 = 256 * 5^6
So, 5^n > 256 * 5^6
5^4 = 625, which is greater than 256 and 5^3 = 125 < 256
So, 5^n should at least be equal to 5^4 * 5^6 = 5^10 for 5^n > 256 * 5^6 to hold true.

Therefore, the least possible value of n is 10.

The correct answer is D.

[GMATGuruNY]

If n is an integer, and 5^n > 4,000,000 then what is the least possible value of n ?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 11

Plug in the answers and compare POWERS OF 10.
Note the following:
4,000,000 = (2²)(10^6)
2^10 = 1024 ≈ 10³.

Answer choice C: n=9
5^9 > (2²)(10^6)
(5^9)(2^9) > (2²)(2^9)(10^6)
10^9 > (2)(2^10)(10^6)
10^9 > (2)(10³)(10^6)
10^9 > (2)(10^9)
The lefthand side -- and thus the value of n -- needs to be just a bit bigger.

The correct answer is D.