# CAT Questions | Number System Questions for CAT

### CAT Verbal Ability

Number System is an important topic that is covered all the time in MBA exams. Students can expect 1-4 questions from Number systems in CAT while in 2-5 can be expected. The difficulty level of the Number systems questions mostly ranges from easy to moderate, but sometimes difficult conceptual questions are also seen in various MBA entrance exams.

## CAT Quantitative Aptitude questions | CAT Questions on Number System

Number system questions that have been asked in CAT previously are as follows: –

1. The number of real-valued solutions of the equation 2x + 2-x = 2 – (x – 2)2 is              [CAT 2020]

A) 1    B) 0     C) Infinite   D) 2

2^x +(1/2^x) =2-(x-2)^2

LHS:

2^x=positive value

2^x +(1/2^x)> =2

RHS:

(x-2)^2>=0 (square of a no. is always positive)

So 2-(x-2)^2<=2

So only possible solution will be when LHS =RHS that is when both of them are equal to 2

2 =2-(x-2)^2 [x=2]

Putting x=2 in LHS

2^2 + (1/2^2) = 4+1/4 [Not =2]

Therefore this Equation has no solution.

2. Let A, B and C be three positive integers such that the sum of A and the mean of B and C is 5. In addition, the sum of B and the mean of A and C is 7. Then the sum of A and B is              [CAT 2020]

A) 7   B) 4       C) 5         D) 6

A+(B+C)/2=5 ->2a+b+c=10

B+(A+C)/2=7 ->2b+a+c=14

subtracting 2 equations we get:

b-a=4 (eliminates option 2 )

3*(a + b)+2c=24

2c will be even so (a + b) also has to be even (eliminates option 3 &4 )

3. How many 3-digit numbers are there, for which the product of their digits is more than 2 but less than 7?  _______________.                                                              [CAT 2020]

Let the 3-digit number be abc so possible values of a×b×c are 3,4,5 and 6.

If a×b×c=3,(a,b,c)=(1,1,3) ,total such numbers=3

If a×b×c=4,(a,b,c)=(1,1,4) or (1,2,2),total such numbers=6

If a×b×c=5,(a,b,c)=(1,1,5) ,total such numbers=3

If a×b×c=6,(a,b,c)=(1,1,6),(1,2,3) ,total such numbers=9

Thus total possible values of abc = 3+6+3+9 = 21

4. The number of integers that satisfy the equality (x2 – 5x + 7)x + 1 = 1 is                      [CAT 2020]

A) 4           B) 2         C) 3       D) 5

Case 1 : if x +1 = 0

Means x = -1

Case 2 :

If x^2-5x+7=1

or x^2-5x+6=0

So x = 2 or 3

Thus, possible number of integer solutions = 3

## List of Concepts in Number System

The list of Number System concepts for solving CAT questions that need to be covered in this chapter are as follows: –

• Algebraic Formulas

• Law of indices

• Binomial Theorem

• Finding out recurring/non-recurring decimal

• Converting recurring/non-recurring decimal to fractions and vice versa – For e.g: – “Convert 0.125125125… to the fractional form”

• Converting mixed recurring decimals to fractions – For e.g: – “Express 0.18888… as a fraction”

• Finding out the last digit/unit’s digit or last two digits– For e.g: – “What is the last digit of A multiplied by B?”

• Finding out the remainder- For e.g: – “Find out the remainder when 7^25 is divided by 6”

• Interchanging of digits – For e.g: – “The digits of a two-digit number are in the ratio of 2 : 3 and the number obtained by interchanging the digits is bigger than the original number by 27. What is the original number?

• Checking the divisibility rules- For 2,3,4,5,6,7,8,9,10,11,13,17,19,23

• Finding out the number of factors- For e.g: – “How many factors of (2^5) * (3^6) * (5^2) are perfect squares?”

• Finding out the number of zeroes – For e.g: – ”What is the number of trailing zeroes in 1123!?”

• Finding out the highest power- For e.g: – “Find the largest power of 3 contained in 95!”

• Base Conversion – From decimal to other bases or vice versa

• Least common multiple (LCM) and Highest Common Factor (HCF) – For e.g – “Find the least number, which is exactly divisible by x,y,z”

• Also, there will be questions from whole numbers, real numbers, natural numbers, composite numbers, rational/irrational numbers, even/odd numbers, and prime numbers

### Some Questions from Previous Papers

Some of the important CAT questions on number system that appeared in the previous papers are: –

1. How many factors of (2^4) *(3^5) *(10^4) are perfect squares which are greater than 1? (CAT 2019 – SLOT 2)

24 x 35 x 104 = 2^4×3^5× (2^4×5^4 ) = 2^8×3^5×5^4
Perfect square factors of the number will be of the form 2^a×3^b×5^c
Where a can be 0, 2, 4, 6 or 8. b can be 0, 2 or 4 and c can be 0, 2 or 4
so total number of square factors = 5*3*3 = 45
But when a=b=c =0 factor will be 1.
Thus, perfect squares which are greater than 1= 45 -1 = 44

2. What is the largest positive integer n such that ( (n^2) + 7n +12)/( (n^2)-n-12) is also a positive integer? (CAT 2019 – SLOT 2)

Given,
(n^2+ 7n+12)/(n^2-n-12 )=(n^2+ 3n+4n+12)/(n^2-4n +3n-12)
=((n+3)(n+4))/(n-4)(n+3)
=(n+4)/(n-4) is an integer
From the option we can see largest possible value of n =12

3.Let a,b,x,y be real numbers such that a^2 + b^2 = 25, x^2 + y^2=169, and ax+by=65. If k=ay-bx, then k=? (CAT 2019 – SLOT 2)

As a,b, x , y are real and as we know 3^2 + 4^2 = 25
Or 5^2 + 0= 25 also 13^2 + 0= 169 and 5^2 + 12^2 = 169
ax+ by = 65 is possible only when (a,b) = (0,5) and (x,y)= (0,13)
Thus k = 0*13 – 0*5 =0

4. The number of integers x such that 0.25 < 2^x < 200, and 2^x +2 is perfectly divisible by either 3 or 4, is? (CAT 2018 – SLOT 1)

As given 0.25 < 2x < 200
Or ¼ < 2^x < 200
So x = { -2, -1, 0,1,2,3,4,5,6,7}
Now 2^x +2 will be perfectly divisible by 3 if x is even non-negative integer and will be divisible by 4 if x = 1
So number of possible solution = 5 { x = 0, 1, 2, 4 ,6}

5. How many numbers with two or more digits can be formed with the digits 1,2,3,4,5,6,7,8,9, so that in every such number, each digit is used at most once and the digits appear in the ascending order? (CAT 2018 – SLOT 1)

As the digits appear in ascending order in the numbers, number of ways of forming a n-digit number using the 9 digits
= 9Cn
Number of possible two-digit numbers which can be formed = 9C2+9C3+9C4+9C5+9C6+9C7+9C8+9C9
=(2^9)−(9C1+9C1)=(2^9)−(9C1+9C1)
=512−(1+9)=502

## How to deal with Numbers preparation

Level 1

• Students are supposed to have basic knowledge of the properties of numbers. You can go through the basic concepts through MBAP CAT E Book (Concept Theory). You can also listen to Live Lecture Recording (Basic) on basic concept and MBAP lecture Assignment.

• They should have a fast calculation mindset.

• Students need to have the skill to identify which type of number system is being used and find solutions. Learn different concepts and number system formula.

• Students need to avoid their calculators and practice calculations using hand and develop speed. You can choose MBAP CAT E Book (Practice Questions) and MBAP CAT Advance E books to practice questions on Number system for CAT.

Level 2

• Topics in Number System such as highest power, last digit, number of factors, interchanging of digits are important and students need to be thorough with these chapters by using MBAP Topic wise Previous Year CAT Questions.

• Number Questions from remainders can be a little twisted, so advising students to practice this chapter carefully. Also, spending too much time is harmful, so try to answer questions that are more straightforward and keep twisted questions to the end

Level 3

Students need to look into number system problems that use multiple concepts to find a solution. Try to answer questions from the advanced level of previous CAT papers, mock tests, and materials. You can choose MBAP Previous Year CAT paper for the same. And, try to attempt these questions in a time-based manner. Taking too much time for advanced questions will be harmful to crack the exam.

Finally, students need to practice as many questions as they can to get a hold of the concepts and find solutions fast. Practice will always help the student to solve CAT questions correctly and efficiently and ultimately crack the exam.

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