5^n>4,000,000

5^n>4*10^6

5^n>4*2^6*5^6

5^n>2^8^5^6

If we sub n=6 RHS contains extra term 2^8.


We need to find the power of n such that 5^n>256.n=4

-->n=10

Pick D

Hey Selango,

Wow...really smart way to do this one. With exponent problems, it almost never hurts to break numbers down into prime factors! Really impressive.

I did it fairly quickly a different (but less efficient) way, so in case it's helpful to anyone here I'll post some strategy that I used.

Knowing that the answer choices were all sequential, I figured I'd get a number in the range of 5^7 and then work from there...I might get lucky with the first one, or outside of that I could "divide and conquer" the answer choices by pinning down values - if 5^7 was too small, then I could try 5^9. If that was too big, it would have to be 5^8; if not, I could multiply by another 25 to try 5^11 and repeat that same process.

Here's my logic:

1) The sequential answer choices mean that there's not any repetitive work...calculating 5^7 will take a second, but then each extra answer choice is just *5.

2) Multiplying by 25 should be pretty easy - it's essentially 1/4 * 100, so you take a quarter of what you have and tack on two zeroes. If you live in an apartment building and have to use quarters for laundry you should already be pretty quick at calculating using quarters!

3) It's a little "brute force" with the calculation, but since multiplying by 25 is so easy I can commit only to having to do at most 3 math problems, which I should be able to do in less than 2 minutes.

So here's what I did:

5^7 = 5^4 * 5^3, so I multiplied 625 by 125.

625 * 100 = 62500 and 625 * 25 is 1/4 of that, so it's just over 15000, so let's estimate it at around 80000 for 5^7.

Clearly 5^7 is too small at 80000, so I'd multiply by 25 to get to 5^9. Again, multiplying by 25 is the same as taking 1/4 and adding two zeroes, so it would be 20000 with two zeroes, or 2,000,000.

5^9 is too small, but at 2,000,000, multiplying by 5 clearly would more than double that number, and since 4,000,000 is what we need to exceed, we can be comfortable calling it 5^10.


Like I said, Selango's method is a little bit more universal, so if you see it his way definitely do that! But for strategic value here, if you peek at the answer choices and see as I did that you actually don't have to do very much work to perform calculations, my method took me well under two minutes. My takeaways for future problems would be:

1) Use the answer choices to your advantage; in this case, recognizing that the answer choices each don't require that much work allows you to "brute force" the problem without taking too much time

2) Get really comfortable multiplying by 25...it should become second-nature if you do it a handful of times (and replicate it in real life when using quarters for laundry, parking meters, etc.) and can save you a lot of calculation time.