1 = [3n + 1] means remainder when
$$7\ \left[3n\ +\ 1\right]\ =\ \frac{21\ +\ 7}{3},\ we\ get$$
remainder as 1 , hence [3n + 1 ] = 1

2 = [3n] means remainder when 7 [3n]
= 21n is divided by 3, we get remainder as 0, hence [3n] = 0

3] = [3n] + 2 means remainder when
$$7\ \left[3n\right]+2\ =\ \frac{21n\ +\ 2}{3},\ we\ get\ remainder\ $$
as 2 hence, [3n] + 2
so the expression in 1 yields a value of 1
Therefore, Answer is option A

I. [3n+1] means remainder when 7(3n+1) = 21n+7 is divided by 3, we get remainder as 1. Hence [3n+1] = 1.

II. [3n] means remainder when 7(3n) = 21n is divided by 3, we get remainder as 0. Hence [3n] = 2.

III. [3n]+2 means remainder when 7(3n)+2 = 21n + 2 is divided by 3, we get remainder as 2. Hence [3n]+2 = 2.

Hence, only I gives remainder as 1. The correct answer is A.

=>

Statement I
7(3n+1) = 21n + 7 = 3(7n+2) + 1
Thus, [3n+1] = 1

Statement II
7(3n) = 21n = 3*7n + 0
Thus, [3n] = 0

Statement III
Since [3n] = 0, we have [3n] + 2 = 2.

Thus, only [3n+1] equals 1.

Therefore, the answer is A.

Answer: A

Max@Math Revolution wrote:[GMAT math practice question]

For a positive integer m, [m] is defined to be the remainder when 7m is divided by 3. If n is a positive integer, which of the following are equal to 1?

I. [3n+1]
II. [3n]
III. [3n] + 2

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

Let's analyze each Roman numeral.

I. [3n+1]

[3n+1] is the remainder when 7(3n + 1) = 21n + 7 is divided by 3. Since 21n is divisible by 3, we see that [3n+1] is equal to the remainder when 7 is divided by 3, which is 1. So I is true.

II. [3n]

[3n] is the remainder when 7(3n) = 21n is divided by 3. Since 21n is divisible by 3, we see that the remainder is 0. So II is not true.

III. [3n] + 2

We saw that in Roman numeral II, [3n] has a remainder 0 when it's divided by 3. So [3n] + 2 has a remainder of 2 when it's divided by 3. III is not true.

Answer: A