Number System Actual Questions in XAT 2023

Q1. The addition of 7 distinct positive integers is 1740. What is the largest possible “greatest common divisor” of these 7 distinct positive integers?

1. 42
2. 60
3. 74
4. 140
5. None of the above.

EXPLANATION

Let the GCD of the 7 different positive numbers be x.
Then the sum of these 7 integers is at least 28x.

The positive integers could be x, 2x, 3x, 4x, 5x, 6x, 7x

So, the maximum value of the GCD occurs when N is as close to 28 as possible.
(1740 is not a multiple of 28.)
29 is a multiple of 1740.
When N = 29, y = 60
So, the largest GCD of these 7 distinct positive integers is 60.

Q2. Five students appeared for an examination. The average mark obtained by these five students is 40. The maximum mark of the examination is 100, and each of the five students scored more than 10 marks. However, none of them scored exactly 40 marks. Based on the information given, which of the following MUST BE true?

1. At least, three of them scored a maximum of 40 marks
2. At least, three of them scored more than 40 marks
3. At least, one of them scored exactly 41 marks
4. At most, two of them scored more than 40 marks
5. At least, one of them scored less than 40 marks

EXPLANATION

Since none of the students scored exactly 40 marks,
at least one of them should score more than 40 marks and
at least one of them should score less than 40 marks.

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The answer is ‘At least, one of them scored less than 40 marks’

Choice E is the correct answer.

Q3. There are three sections in a question paper and each section has 10 questions. First section only has multiple-choice questions, and 2 marks will be awarded for each correct answer. For each wrong answer, 0.5 marks will be deducted. Any un-attempted question in this section will be treated as a wrong answer. Each question in the second section carries 3 marks, whereas each question in the third section carries 5 marks. For any wrong answer or un-attempted question in the second and third sections, no marks will be deducted. A student’s score is the addition of marks obtained in all the three sections. What is the sixth highest possible score?

1. 92.5
2. 94
3. 95.5
4. 95
5. None of the above.

EXPLANATION

Let’s imagine a scenario where the student scores all the questions correctly.

The maximum possible score is 2×10+3×10+5×10=100.

For each mistake/non-attempt, you lose 2 + 0.5 marks. In the second and third sections, you lose 3 and 5 marks respectively.