Number property: how to ensure that 5% greater is an integer

[serendipiteez]

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

Answer D

Why do we look at b from the perspective of a? Once you get [spoiler]a = 20/21b or b = (21/20)*a[/spoiler]. Why cant we plug in the values of b into I, II and III above?

[mkdureja]

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

Answer D

Why do we look at b from the perspective of a? Once you get [spoiler]a = 20/21b or b = (21/20)*a[/spoiler]. Why cant we plug in the values of b into I, II and III above?

As a and b both have to be integers.
When you get to the point, b=(21/20)*a
That means for 'b' to be an integer 'a' must be divisible by 20.
That makes it for 'b' to be divisible by 21, or both 3 and 7.

[GMATGuruNY]

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

Answer D

b = a + (5/100)a = (105/100)a = (21/20)a.
Thus:
b/a = 21/20.

Since 21/20 cannot be further reduced, and b and a are positive integers, b must be a multiple of 21, while a must be a multiple of 20:
b=21 and a=20, so that b/a = 21/20.
b=42 and a=40, so that b/a = 42/40 = 21/20.
b=63 and a=60, so that b/a = 63/60 = 21/20.
And so on.

In every case, b/3 and b/7 will be equal to integer values.
Thus, the correct answer choice must include statements I and III.

The correct answer is D.