If a and b are positive integers, what is the greatest possible value

Question

500^{ab}<2^{100}

If a and b are positive integers, what is the greatest possible value of a?

(A) 2
(B) 5
(C) 10
(D) 11
(E) 13

Luckisnoexcuse · Accepted Answer

500^ab < 512^100/9

since ab = ~11.11 and minimum value of b will be 1
a can take highest value of 11
D

pushpitkc · Answer

Given data : 500^{ab}<2^{100} We have been asked to find out the greatest possible value for a. 500 when prime factorized gives 2^2 * 5^3 Therefore, 500^{ab} = 2^{2ab} * 5^{3ab} For the maximum possible value of a, we need to have the lowest value for b. The lowest positive value for b = 1. The expression now becomes 2^{2a} * 5^{3a} < 2^{100} Now going by answer options, If we go by answer option E, a=13 : 2^{26} * 5^{39} < 2^{100} => 5^{39} < \frac{2^{100}}{2^{26}} 5^{39} > 2^{74} => (5^2)^{19} *5 > (2^4)^{18} *4 (Extrapolating 5^2 > 2^4) For answer option D, a=11 : 2^{22} * 5^{33} < 2^{100} => 5^{33} < \frac{2^{100}}{2^{22}} 5^{33} < 2^{78} => (5^2)^{16} * 5 < (2^4)^{19} * 4 (Extrapolating 5^2 > 2^4) Hence, a=11 is the highest value of a possible such that the expression holds true(Option D)

kumarparitosh123 · Answer

Imo B.
Solving we get 
2^(2*a*b) X 5^(3*a*b) < 2^100.
To get maximum value of A we need to minimise B. Thus the least value that can b assigned to B is 1.
So after putting different values I found that the maximum value of A to be 5 because if we put A as 10 then the left hand side becomes greater than right hand side.
So I will go for B.

# Bunuel if I have missed something, kindly correct me. 
Thanks in advance.

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kumarparitosh123 · Answer

Kudos !! 
Great solution. Now I understand my mistake. Thanks for such good illustration.

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zanaik89 · Answer

what do u mean by extrapolating?

pls explain..thanks

zanaik89 · Answer

Howdid u get 512^100/9?
Pls explain thanks

Luckisnoexcuse · Answer

2^9 = 512

ScottTargetTestPrep · Answer

Since we have to maximize the value of a, we must minimize the value of b. Since b is a positive integer, the smallest value of b is 1. Thus, letting b = 1, we have:

500^a < 2^100

Since 2^9 = 512, which is slightly larger than 500, we can say:

500^a < (2^9)^(100/9)

500^a < 512^(100/9)

Since 500 < 512, we see that if a ≤ 100/9 = 11.11, then 500^a < 2^100. Therefore, so far we can say the maximum value of a is 11. However, can a be 13 and 500^a < 2^100 still be true? Let’s prove (or disprove) it:

If a = 13, then 500^13 = (2^2 x 5^3)^13 = 2^26 x 5^39, so the question becomes:

Is 2^26 x 5^39 < 2^100?

Is 5^39 < 2^74?

The answer is no, since 5^39 > 4^39 = (2^2)^39 = 2^78. Since 5^39 is greater than 2^78, so it must be greater (not less) than 2^74. So, we can see that a can’t be 13. Thus, the largest value of a must be 11.

Answer: D

syedazeem3 · Answer

500^a < (2^9)^(100/9)

500^a < 512^(100/9)

I get how you got to 512, but can you please explain why you divided 100 by 9 specifically? I’m a bit confused on why used 9..

Thanks again

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pushpitkc · Answer

syedazeem3

This is a very important formula that deals with exponents:  (a^m)^n = a^{m*n}

Going by that,  2^{100} = (2^9)^{\frac{100}{9}}  because  100 = 9*\frac{100}{9}

Apologies for the delay in clearing your confusion. Hope that helps you!

spc11 · Answer

Here is how I approach this, experts please let me know if this isn't the right approach:

500^ab = (5^3 * 2^2)^ab
 = 5^3ab * 2^2ab
 ~ (2^2)^3ab * 2^2ab - Taking 5 ~ 4 = 2^2
 = 2^6ab *2^2ab
 = 2^8ab

Now, 2^8ab < 2^100

that is : 8ab < 100
or, ab < 12.5

taking b =1 as its least value,
greatest value of a will be one nearest to 12.5 ie 11, since we have approximated 5 to 4 and hence a little loss of value of a gets compensated.

And, eliminating the options will also give us 11 as the greatest possible value of a.
Option D

Hope this helps!

cbh · Answer

500^ab<2^100
(2^2ab)*(5^3ab)<2^100
taking natural log
ln((2^2ab)*(5^3ab))<ln(2^100)
2ab*ln(2) + 3ab*ln(5)<100*ln(2) -> this is a standard log property, actually 2 of them, log (5*5) can be written as log5 +log5 and log(5^5) can be written as 5*log5 (5 times log of 5)

ln 2 ~ 0.7
ln 5 ~ 1.6
b=1 (when a is maximum)

2a*0.7+3a*1.6<70
6.2a<70
only one answer matched 11

Kinshook · Answer

Given:  500^{ab}<2^{100}

Asked: If a and b are positive integers, what is the greatest possible value of a?

500^{ab} < 2^{100} = 2*512^{11}
Greatest value of a is when b=1
Greatest value of a = 11 Since
500^{11} < 2*512^{11}
But
500^{12} = 500*500^{11} > 2*512^{11}

IMO D

Fdambro294 · Answer

Same type of Concept, except I used a bit bigger Bases to Break Down the 2's and 5's:

(2)^9 = 512

(5)^4 = 625

I felt like you could see which answer choices clearly would not work a bit easier.

Again, start with the Idea that in order to Maximize the Power = a, we need to Minimize the Power = b. Since they are + Integers, b = 1.

(500)^a < (2)^100

(2)^2a * (5)^3a < (2)^100

Starting with Answer Choice E - Can the exponent a = 13 and the Inequality Still Stand as True?

(2)^26 * (5)^39 < (2)^100 ?

(5)^39 < (2)^100 / (2)^26 ?

(5)^39 < (2)^74

Now Pulling out 4 Powers of 5 to make ONE Base = 625 ----

and Pulling out 9 Powers of 2 to make ONE Base = 512 ----

(625)^9 * (5)^3 < (512)^8 * (2)^2 ?

(625)^9 * 125 < (512)^8 * 4 ?

on the Left Side ---- you have a Larger Base, Lager Exponent, and * 125 Multiplying all of it compared to the Right Side.

Therefore, a can NOT = 13

-D- can a = 11?

again, Minimizing the Power b = +1

(500)^a < (2)^100

(500)^a <   ^11 * 2

(500)^a < (512)^11 * 2

if a = 11, we are raising a SMALLER Base of 500 to the Same Power of 11 than we are Raising a HIGHER Base of 512 too. Therefore, we know for sure that

(500)^11 < (512)^11 * 2

Since D is the Highest Answer Choice that Works, it must be D - 11

ishita27 · Answer

IT IS NOT A 95% difficulty level question if you follow the correct logic

500^ab < 2^100

(5*2*5*2*5)^ab < 2^100

We can say 5*5*5 = 125 which is approximately equal to 2^7 = 128
So,
(2^7*2*2)^ab < 2^100
2^9ab < 2^100
9ab < 100
ab < 11.11

Therefore greatest possible value of a = 11, where b=1

Correct Answer is D

Fdambro294 · Answer

Great logic. Loved the explanation. Very quick and easy.

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Wadree · Answer

500^ab < 2^100 ..... and we gotta make ...a.... as big as we can....so we gotta make.....b.....as small as we can....but ...a....
and....b....are positive integers....so the smallest that....b.... can be is 1.....
So..... 500^a < 2^100
Now.... 2^8 = 256 ≈ 500 ≈ 512 = 2^9 
If we take 500 = 2^8 .....then.....2^8a < 2^100.....so highest value of...a... would be 11 among da options......
Because.....if a = 11 then... 2^88 < 2^100....
And...if we take 500 = 2^9....then....2^9a < 2^100....so highest value of...a... would still be 11....
Because.....if a = 11 then... 2^99 < 2^100 ....
So....500 is between 2^8 and 2^9.....so if we take 500 = 500...still then highest value of...a... would be 11.....bingo.....

If a and b are positive integers, what is the greatest possible value

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If a and b are positive integers, what is the greatest possible value

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