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If the positive integer b is 5% greater than the positive in

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HarveyS
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If the positive integer b is 5% greater than the positive in [#permalink] New post   Updated on: 12 Jan 2014, 19:40
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If the positive integer b is 5% greater than the positive integer a, which of the following expressions must be equal to an integer?

I. b/3
II. b/5
III. b/7

A. None
B. II only
C. I and II only
D. I and III only
E. II and III only
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D

Originally posted by HarveyS on 12 Jan 2014, 16:53.
Last edited by HarveyS on 12 Jan 2014, 19:40, edited 2 times in total.
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Bunuel
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Ratio Question [#permalink] New post   03 Aug 2014, 19:22
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If the positive integer b is 5% greater than the positive integer a, which of the following expressions must be equal to an integer?

I. b/3
II. b/5
III. b/7

A. None
B. II only
C. I and II only
D. I and III only
E. II and III only

The positive integer b is 5% greater than the positive integer a: \(b = 1.05a\);

\(b = \frac{21}{20}a\);

\(\frac{b}{a} = \frac{21}{20}\);

b is a multiple of 21 (and a is a multiple of 20).

Therefore b/3 and b/7 will be equal to an integer.

Answer: D.
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Re: If the positive integer b is 5% greater than the positive in [#permalink] New post   12 Jan 2014, 18:09
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I would pick D

b=a(1+(5/100))
b=a+(a/20)

looks like b will be comprised of a positive multiple of 20 + the multiple

b = a + (20a)
b=21a

I believe this makes b positive multiple of 21, so it will have a (7)(3) in its factors.
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Re: If the positive integer b is 5% greater than the positive in [#permalink] New post   25 Jun 2016, 08:23
Bunuel wrote:
Mountain14 wrote:
If the positive integer b is 5% greater than the positive integer a, which of the following expressions must be equal to an integer?

I. b/3
II. b/5
III. b/7

A.None
B.II only
C.I and II only
D.I and III only
E.II and III only


OA after some discussion


The positive integer b is 5% greater than the positive integer a --> \(b=1.05a\) --> \(b=\frac{21}{20}a\) --> \(\frac{b}{a}=\frac{21}{20}\) --> b is a multiple of 21 (and a is a multiple of 20). Therefore b/3 and b/7 will be equal to an integer.

Answer: D.

P.S. OA must be indicated from the beginning.


Hi Bunuel,

I have a doubt, though I understood the process. I prefer working with numbers instead of using equations.

So for this question my approach was to work with distant numbers

i.e a=10 and b=10.5 - 1
and a=100 b =105 - 2

In case 1, b/3 or b/5 or b/7 satisfies.
In case 2, All three do.

As this was an always true question, I went with Choice A i.e None.

Why does working with numbers does not work here, and how to distinguish questions where we should and should not use numbers.

Your guidance will really be appreciated.

TIA :)
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Re: If the positive integer b is 5% greater than the positive in [#permalink] New post   25 Jun 2016, 09:26
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Mountain14 wrote:
If the positive integer b is 5% greater than the positive integer a, which of the following expressions must be equal to an integer?

I. b/3
II. b/5
III. b/7

A. None
B. II only
C. I and II only
D. I and III only
E. II and III only


Plug in some number -

a = 100 , b = 105

Here all the options ( I ) , ( II ) & ( III ) will produce integers

Plug in some diff number -

a = 120, b = 126

Here we can get an integer with options ( I ) & ( III )

Plug in some diff number -

a = 140 , b = 147

Here we can get an integer with options ( I ) & ( III )

So you find options ( I ) & ( III ) common in all the values, so answer must be (D)


believer700 wrote:
Why does working with numbers does not work here, and how to distinguish questions where we should and should not use numbers.


Working with numbers works fine here but you will need to try different values before concluding an answer...

Hope this helps !!
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Re: If the positive integer b is 5% greater than the positive in [#permalink] New post   26 Jun 2016, 00:48
The smallest a has to be 20, in which case b is 21. 20 is the smallest value of a because b-a, or 5% of a, has to be a positive integer, and the smallest value of a positive integer is 1.

B is obviously divisible by 3 and 7 but not 5.

Answer choice D.
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Re: If the positive integer b is 5% greater than the positive in [#permalink] New post   09 May 2022, 05:03
b=a+0.05a
For b to be an integer 0.05a must be an integer too. Hence any multiple of 0.05a must be an integer too.
Now,
b/3=0.35a (a multiple of 0.05a, hence an integer)
b/5=0.21a (not a multiple of 0.05a, hence cannot be an integer)
b/7=0.15a (a multiple of 0.05a, hence an integer)

Thus answer is “D”

Posted from my mobile device
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Re: If the positive integer b is 5% greater than the positive in [#permalink] New post   11 Jun 2023, 23:45
Correct me If i am wrong, In a 'Must Be' type question why is everyone generalizing integer is greater than 21. Question doesnt hold true for Integers 1-20 and question doesn't explicitly imply integer is greater than 21.

I chose A on that grounds and skipped D.

Bunuel ScottTargetTestPrep KarishmaB your opinions would be helpful on this subject.
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Re: If the positive integer b is 5% greater than the positive in [#permalink] New post   11 Jun 2023, 23:52
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RudreshKal wrote:
Correct me If i am wrong, In a 'Must Be' type question why is everyone generalizing integer is greater than 21. Question doesnt hold true for Integers 1-20 and question doesn't explicitly imply integer is greater than 21.

I chose A on that grounds and skipped D.

Bunuel ScottTargetTestPrep KarishmaB your opinions would be helpful on this subject.


I suggest to re-read the solution here.

In short, we get that b is a positive multiple of 21, so it can be 21, 42, 63, and so on (the least value of b is 21). If a number is a multiple of 21, then it's a multiple of both 3 and 7 too, thus both b/3 and b/7 will be integers.
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Re: If the positive integer b is 5% greater than the positive in [#permalink] New post   12 Jun 2023, 04:54
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RudreshKal wrote:
Correct me If i am wrong, In a 'Must Be' type question why is everyone generalizing integer is greater than 21. Question doesnt hold true for Integers 1-20 and question doesn't explicitly imply integer is greater than 21.

I chose A on that grounds and skipped D.

Bunuel ScottTargetTestPrep KarishmaB your opinions would be helpful on this subject.


I have a positive integer a. When I increase it by 5%, I get another positive integer b. Think of some values that a and b can take.

If a = 1 (smallest positive integer), b will be 1.05 (not an integer)
If a = 2, b will be 2.1 (not an integer)
and so on...
If a = 20, then b will be 21 (the first pair in which both are integers)
If a = 21, then b will be 22.something (not an integer)
and so on..
Next when a = 40, b will be 42 (both integers)

Basically values of a will be multiples of 20 and values of b will be multiples of 21.
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Re: If the positive integer b is 5% greater than the positive in [#permalink]
12 Jun 2023, 04:54
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