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

Question

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

Bunuel · Accepted Answer

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.

m3equals333 · Answer

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.

believer700 · Answer

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

Abhishek009 · Answer

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)

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

HanoiGMATtutor · Answer

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.

ankur9589 · Answer

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”

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RudreshKal · Answer

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 chose A on that grounds and skipped D.

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

Bunuel · Answer

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.

KarishmaB · Answer

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.

Capablanca · Answer

From the statement, we have: b = 1.05a

So b/(1.05) = a, which is an integer.

b/(21/20) = integer

20b/21 = integer

Since 20 does not have prime factors 3 and 7, b must be divisible by 3 and 7 to get an integer.

Answer (D)

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

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