Problem Solving

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

39
30
28
27
17

Pls post your answers with explanations.

cvs wrote:Which of the following best approximates the value of q if 5^28+3^11=5^q ?

39
30
28
27
17

Pls post your answers with explanations.

The answer is B 30.

First the basics .. 5^n when divided by 3 has a remainder 1 if n is even and a remainder 2 if n is odd ..

Now, the given equation can also be written as 3^11=5^q-5^28, looking at the equation we can say that 5^q-5^28 has to be a multiple of 3(that means when (5^q-5^28) is divided by 3 the remainder is 0) that means 5^q and 5^28 should have similar remainders when divided by 3, so that when we subtract them they cancel out and the overall remainder is 0.

Now, we know that 5^28 has a remainder 1 when divided by 3 (as 28 is even), so 5^q should also have a remainder 1 when divided by 3 and hence we can say that q should also be even. The given answer choices have just 2 even options 30 and 28, obviously 28 cant be the answer therefore we can conclude q has to be 30.

Regards

Any more thoughts.. The answer is not 30!!!

IMO 28

now 5^28 would be a huge nos itself

& 3^11 = 27*27*27 * 9

to approx we can say it is 30^3 *10 which is 180000

this when added to such a huge nos as 5^28 would not make much of a diff

so q =28

samirpandeyit62 wrote:IMO 28

now 5^28 would be a huge nos itself

& 3^11 = 27*27*27 * 9

to approx we can say it is 30^3 *10 which is 180000

this when added to such a huge nos as 5^28 would not make much of a diff

so q =28

Well I had thot about it ... but if we choose q = 28 then we are considering the value of 3^11 as 0, which does not make a lot of sense to me .. but then again the question does say approximate value of q, but dont u think we would be pushing it a bit by assuming 3^11 equal to 0 ..

On second thought I agree the answer should be 28 .. I was sure that the answer should be either 30 or 28, 28 makes more sense now .. good going Samir

Regards

Hi Gabriel,
My reasoning was like this

5^28 = very large nos say X

3^11 = 180000 approx

now q can be 28 or more, i.e next nos is 30

so if we line them up it would be

X X+180000 25X

now obviously X+180000 would be closer to X

Thanks Samir/Gabrial.

The official answer is 28.

Samir explanation is identical to MGMAT's explanation.

Hi samirpandeyit62,

I don't understand this approximation:
3^11 = 27*27*27 * 9
to approx we can say it is 30^3 *10 which is 180000

Also, can you clarify on this as well:

5^28 = very large nos say X
3^11 = 180000 approx
now q can be 28 or more, i.e next nos is 30
so if we line them up it would be
X X+180000 25X
now obviously X+180000 would be closer to X

Also, I see what you wrote before that adding 3^11 to 5^28 will not make a major difference, so then why can't Q be 30.....to account for the slight increase when we add to 5^28?

Thanks