Exponent - Number theory Question Bank

[shaheryarmian]

Q) Which of the following best approximates the value of q if 5^28+3^11=5^q ?

39
30
28
27
17

For some reason the answer was 28 for this question. I tried to solve it as below so having a problem with 28 as an answer:

3^11 = 5^q - 5^28
therefore is q = 28 then the 2 sides cant be equal because the right side would = 0. Am I doing something wrong here?

[StaceyKoprince]

Please read (and follow!) the forum guidelines before posting. Problems should not be posted in this folder; this is for general strategy questions only. Problem moved to MGMAT Quant Strategy Guide folder.

[sarathy.srinivas]

Hi, you seem to have misunderstood/misread the question. The question is not asking for the exact value, it is asking for the best answer.

You have arrived at

3^11 = 5^q - 5^28

instead of
3^11 ~ 5^q - 5^28
which can be further simplied to
3^11 ~ 5^28[(5^(q-28))-1]

Now, by examining the answer choices, we can eliminate 39, 27 and 17. Which leaves us with 28 and 30. This is where things get tricky. I had some trouble with this, but after a bit of soul searching (read head scratching), common sense took over...
Substitute the two answer choices in the simplified equation and we will be left with
3^11 ~ 0
AND
3^11 ~ (5^28)*((5^2)-1) which can be simplified to
3^11 ~ (5^28)*24

3^11 is closer to 0 than to the huge number we'd get with putting q = 30.

Therefore, the answer (that q best approximates to) is 28.


Note: This is my first attempt at answering/ explaining a problem on this forum (or any forum for that matter). Please forgive my mistakes, if any.

[Ben Ku]

Sarathy makes a good point. We aren't looking for what q is exactly; we're just looking for what it's closest to.

Let's suppose we had a different question:
10^28 + 5^11 = 10^p

Because 10^28 is sooo much bigger than 5^11, the 5^11 is negligible. Adding 5^11 will not even get it close to 10^29. In fact, even if it were 10^28 + 10^28, you still wouldn't get it to 10^29. So in this alternate question, p is close to 28, because the 5^11 has a negligible effect on the sum.

In this question, 5^28+3^11=5^q, 3^11 is very small compared to 5^28, so q would still be 28.

[shaheryarmian]

thanks guys....i gues I went about trying to prove it algebraically so got confused but the whole point about estimation makes sense.

[Ben Ku]

i'm glad that helped.