Can someone help me figure out which statement must be true here?

Question

When 24 is divided by the positive integer n, the remainder is 4. Which of the following statements about n must be true? 
I. n is even. 
II. n is a multiple of 5.
III. n is a factor of 20

A. III only 
B. I and II only 
C. I and III  only 
D. II and III only 
E. I, II, and II

Jayant M · Accepted Answer

If dividing 24 by n leaves a remainder of 4, that means 24 = qn + 4 for some integer q. Rearranging, you get 20 = qn. So n must be a factor of 20. But it also needs to be less than 24 and greater than 4 (since remainder must be less than n). That leaves n = 5, 10, or 20.
Now go through the statements. I is not necessarily true , n could be 5, which is odd. So eliminate I. II is true , all possible n values (5, 10, 20) are multiples of 5. III is also true,  5, 10, and 20 are all factors of 20.
That means only II and III must be true. Final answer is D.

B.K Sharma · Answer

Here&rsquo;s how to solve it  logically and quickly:

The condition means 24 = qn + 4,  so 20 = qn
That makes n a factor of 20
Possible values of n are 5, 10, or 20 (must be > 4 and divide 20)
I is not always true &mdash; 5 is allowed and it&rsquo;s odd
II is true &mdash; 5, 10, and 20 are all multiples of 5
III is true &mdash; all possible n values divide 20 exactly

Only II and III must be true. Answer is D.