GMAT Practice Question: When 24 is divided by the positive integer n...
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 .
- III only
- I and II only
- I and III only
- II and III only
- I, II, and III
Topics: number theory, divisibility, remainders, must be true statements
Solution
Step 1
Express the division of 24 by n with remainder 4 using the division statement.
24 = n \times q + 4
Step 2
Subtract the remainder from 24 to find a multiple of n.
24 - 4 = n \times q
Step 3
Simplify to see that n divides 20.
20 = n \times q
Step 4
List the positive divisors of 20 that are greater than the remainder (4).
\text{Factors of 20} = 1,2,4,5,10,20 \text{Possible values for }n: 5, 10, 20
Step 5
Check statement I: Is n always even?
n = 5, 10, 20 5 is odd, so I is false.
Step 6
Check statement II: Is n always a multiple of 5?
n = 5, 10, 20 5, 10, 20 are all multiples of 5, so II is true.
Step 7
Check statement III: Is n always a factor of 20?
n = 5, 10, 20 5, 10, 20 are all factors of 20, so III is true.
Answer
D. II and III only
