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PQID = DS49502.01 | OG 2020: Question No. 318

If x is an integer greater than 0, what is the remainder when x is divided by 4?

1. The remainder is 3 when x + 1 is divided by 4.
2. The remainder is 0 when 2x is divided by 4.

Source	OG 2020
PQID	DS49502.01
Type	Data Sufficiency
Topic	Number Properties
Sub-Topic	Remainders
Difficulty	Medium

Solution

Steps 1 & 2: Understand Question and Draw Inferences

In this question, we are given

• The number x is an integer, and x > 0

We need to determine

• The remainder, when x is divided by 4.

To find the remainder, when x is divided by 4, we need to know either the exact value of x, or the general expression expressing the number x.

As we do not have any information present in the question statement, let us now analyse the individual statements.

Step 3: Analyse Statement 1

As per the information given in statement 1, when x + 1 is divided by 4, the remainder is 3.

• Hence, we can say x + 1 = 4 × k + 3, where k is a non-negative integer.
  • x = 4 × k + 2
  • By comparing the above equation with dividend = divisor × quotient + remainder, we can say that the remainder is 2.

Therefore, statement 1 is sufficient to answer the question.

Step 4: Analyse Statement 2

As per the information given in statement 1, when 2x is divided by 4, the remainder is 0.

• Therefore, 2x is a multiple of 4.
• This can happen in two cases:
  • When x is already a multiple of by 4
    • x = 4 × k + 4 = 4(k + 1)
    • Then 2x will also be divisible by 4.
  • When x is not a multiple of 4 but x is a multiple of 2.
    • Hence, x = 4 × k + 2 = 2(2k +1)
      • And, 2x =4 (2k + 1) which makes 2x a multiple of 4.

Therefore, from this statement the remainder can be either 0 or 2.

Hence, statement 2 is not sufficient to answer the question.

Step 5: Combine Both Statements Together (If Needed)

Since we can determine the answer from statement 1 individually, this step is not required.

Hence, the correct answer choice is option A.

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Takeaways

1. Any integer ‘p’ when divided by another integer ‘q’ can be represented in the form: Dividend = Divisor × Quotient + Remainder.
   • So, p = qm + r, where ‘m’ and ‘r’ represent the quotient and the remainder, respectively.
   • Also, r must lie in the range 0 ≤ r < q.