CAT Exam  >  CAT Tests  >  Level-wise Tests for CAT  >  Test Level 3: Number System - 2 - CAT MCQ

Test Level 3: Number System - 2 - CAT MCQ


Test Description

20 Questions MCQ Test Level-wise Tests for CAT - Test Level 3: Number System - 2

Test Level 3: Number System - 2 for CAT 2024 is part of Level-wise Tests for CAT preparation. The Test Level 3: Number System - 2 questions and answers have been prepared according to the CAT exam syllabus.The Test Level 3: Number System - 2 MCQs are made for CAT 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test Level 3: Number System - 2 below.
Solutions of Test Level 3: Number System - 2 questions in English are available as part of our Level-wise Tests for CAT for CAT & Test Level 3: Number System - 2 solutions in Hindi for Level-wise Tests for CAT course. Download more important topics, notes, lectures and mock test series for CAT Exam by signing up for free. Attempt Test Level 3: Number System - 2 | 20 questions in 40 minutes | Mock test for CAT preparation | Free important questions MCQ to study Level-wise Tests for CAT for CAT Exam | Download free PDF with solutions
Test Level 3: Number System - 2 - Question 1

If N = (11111111)2, then what is the sum of the digits of N?

Detailed Solution for Test Level 3: Number System - 2 - Question 1

12 = 1, 112 = 121, 1112 = 12321, 11112 = 1234321 and 111112 = 123454321
⇒ 11111111= 123456787654321
The sum of the digits of given number is 64.

Test Level 3: Number System - 2 - Question 2

A is the smallest integer which when multiplied by 3 gives a number made of 5's only. What is the value of C3, if sum of the digits of A is B and sum of the digits of B is C?

Detailed Solution for Test Level 3: Number System - 2 - Question 2

Smallest number made of 5's and divisible by 3 is 555.
Thus, A = 555/3 = 185
B = 1 + 8 + 5 = 14
C = 1 + 4 = 5
Thus, value of C3 = 53 = 125

1 Crore+ students have signed up on EduRev. Have you? Download the App
Test Level 3: Number System - 2 - Question 3

ab and cd are 2 two-digit natural numbers. Given that 4b + a = 13k1 and 5d - c = 17k2, where k1 and k2 are natural numbers. Find the largest number that will always divide the product of ab and cd.

Detailed Solution for Test Level 3: Number System - 2 - Question 3

ab × cd = (10a + b)(10c + d) 
= (40b + 10a - 39b)(51d - (50d - 10c))
= (10(4b + a) - 39b)(51d - 10(5d - c))
= (10 × 13k1 - 39b)(51d - 10 × 17k2)
= 13 × 17(10k1 - 3b)(3d - 10k2)
= 221(10k1 - 3b)(3d - 10k2)
Hence, the largest number that will divide the product ab and cd is 221.

Test Level 3: Number System - 2 - Question 4

Find the last digit of the following expression.

(15)256 + (19)138 + (32)97

Detailed Solution for Test Level 3: Number System - 2 - Question 4

Unit digit of (15)256 = 5.
As any number with 5 at the units place will always have a 5 at its units place, irrespective of the positive natural number power.
Now, 9= 9, 9= 81, 9= 729, 94 = 6561
So, basically, a number with 9 at its units place raised to an even number, will have 1 at its units place and a number with 9 at its units place raised to an odd number, will have 9 at its units place.
Thus, the units digit of 19138 is a 1.
Now, 2= 2, 22 = 4, 23 = 8, 24 = 16 and 25 = 32
So, basically after every 4th power, the units digit repeats itself.
Now, 97 = 96 + 1 = 24 x 4 + 1
Thus, we have 24 sets of 4 and then 1 more.
So, the number should have 2 at the units place.
Thus, the digit at the units place is 5 + 1 + 2 = 8
Hence, option 1 is correct.

Test Level 3: Number System - 2 - Question 5

Find the value of a in the identity (11100)2 = (1001)a

Detailed Solution for Test Level 3: Number System - 2 - Question 5

 (11100)2 = (28)10
(1001)a = 1 × a3 + 0 × a2 + 0 × a + 1 = a3 + 1 = (1 + a3)
According to the question, (28)10 = (a3 + 1)10
 a3 = 27
 a = 3

Test Level 3: Number System - 2 - Question 6

A number when divided by 296 leaves the remainder 75. If the same number is divided by 37, then what will be the remainder?

Detailed Solution for Test Level 3: Number System - 2 - Question 6

Let x be the number.
Given: x = 296n + 75 [where n is the quotient]
x = 37(8n + 2) + 1
Hence, when x is divided by 37, it gives 1 as the remainder.

Test Level 3: Number System - 2 - Question 7

If N = (7p + 4)(5q)(23) is a perfect cube, where p and q are positive integers, then find the smallest possible value of p + q.

Detailed Solution for Test Level 3: Number System - 2 - Question 7

In order for N to be a perfect cube, each prime factor of N must have the exponent that is divisible by 3.
Since p and q must be positive integers, the smallest value of q and p is 3 and 2, respectively.
Thus, the smallest value of p + q is 5.

Test Level 3: Number System - 2 - Question 8

There are four prime numbers written in ascending order of magnitude. The product of the first three is 385 and that of the last three is 1001. Find the first number.

Detailed Solution for Test Level 3: Number System - 2 - Question 8

The best way is to factorise the number i.e 385 = 5 × 7 × 11
1001 = 7 × 11 × 13
So, the four prime numbers are 5, 7, 11 and 13.
Hence, first prime number = 5

Test Level 3: Number System - 2 - Question 9

What is the greatest positive power of 5 that exactly divides 30!?

Detailed Solution for Test Level 3: Number System - 2 - Question 9

The total number of multiples of 5 in 30! = 6 (5, 10, ….., 30)
The total number of multiples of 5 to the power 2 = 25 in 30! = 1 (25 only)
Further powers of 5 cannot be these as 5 to the power 3 = 125 > 30
Therefore, the greatest power of 5 that divides 30! exactly = 6 + 1 = 7

Test Level 3: Number System - 2 - Question 10

How many multiples of 32 are perfect squares, less than 104?

Detailed Solution for Test Level 3: Number System - 2 - Question 10

The least multiple of 32, which is a perfect square is 32 × 2 = 64.
So, all the numbers of the form 64k (where k is a perfect square) are perfect squares.
i.e. 64 × 12, 64 × 22, 64 × 32, 64 × 42, … 64 × 122
If we take 64 × 132;
64 × 169 > 104
So, total 12 numbers are there.

Test Level 3: Number System - 2 - Question 11

Let A, B and C be positive integers satisfying A < B < C and A + B + C = X. What is the smallest value of X that does not determine A, B and C uniquely?

Detailed Solution for Test Level 3: Number System - 2 - Question 11

We have to check the options one by one.
For X = 9, A + B + C = 9, we have more than one solution, i.e. (1, 2, 6), (1, 3, 5), etc. So, X = 9 does not determine A, B and C uniquely.
For X = 6, we have a unique solution, i.e. (1, 2, 3).
For X = 7, we have a unique solution again, (1, 2, 4).
For X = 8, we have more than one solution, i.e. (1, 2, 5); (1, 3, 4). 
Hence, 9 and 8 are the two values that do not determine A, B and C uniquely. Of the two, 8 is the required smallest value.

Test Level 3: Number System - 2 - Question 12

If, in a number system, 40132 corresponds to 2542 in decimals, find the base of the number system.

Detailed Solution for Test Level 3: Number System - 2 - Question 12

Since 4 is one of the characters in the given number, the base for that number system must be > 5.
Assume that the base is 5. Then, (4 × 54) + (0 × 53) + (1 × 52) + (3 × 51) + (2 × 50) = 2542
So, by estimation, we can say that the base of the number system is 5.

Test Level 3: Number System - 2 - Question 13

Let D be a decimal of the form D = 0.abcdabcdabcd …, where the digits a, b, c and d are integers lying between 0 and 9. At most three of these digits are zero. By what number should D be multiplied so that the result is a natural number?

Detailed Solution for Test Level 3: Number System - 2 - Question 13

D = 0.abcdabcdabcd...
10000D = abcd.abcdabcd...
10000D - D = (abcd.abcdabcd...) - (0.abcdabcdabcd...)
9999D = abcd
D = abcd/9999
From the options, 49995 is the only multiple of 9999.

49995 = 9999 × 5

Test Level 3: Number System - 2 - Question 14

If abc is a three-digit whole number, such that abc = a3 + b3 + c3. [300 < abc < 400], find the sum of a, b and c.

Detailed Solution for Test Level 3: Number System - 2 - Question 14

Since 300 < abc < 400
 100a + 10b + c = a3 + b3 + c3
Put a = 3 and rearrange:
300 + 10b + c = 27 + b3 + c3
 (b3 - 10b) + (c3 - c) = 273
Now, put b = 6 and b = 7 (as b and c are less than equal to 9, and 63 and 73 are the nearest cubes of 273)
For b = 6, we get c3 - c = 117, and no value of c satisfies it.
For b = 7, we get c3 - c = 0, and two values of c, i.e. 0 and 1 satisfy it.
Thus, abc = 370 or 371
Hence, abc cannot be uniquely defined. So, option 4 option is the answer.

Test Level 3: Number System - 2 - Question 15

A number consists of three consecutive digits in order, the one in units place being the greatest of the three. The number formed by reversing the digits exceeds the original number by 22 times the sum of the digits. Find the number.

Detailed Solution for Test Level 3: Number System - 2 - Question 15

Let the digits at hundred's place be x.
Hence the number will have the digits as x, x + 1 and x + 2.
Number will be 100x + 10x + 10 + x + 2 = 111x + 12
Number obtained after reversing the digits = 100x + 200 + 10x + 10 + x = 111x + 210
Sum of the digits = x + x + 1 + x + 2 = 3x + 3
According to the question,
⇒ 111x + 210 = 111x + 12 + 22(3x + 3)
⇒ 210 = 12 + 66x + 66
⇒ 66x = 132
⇒ x = 2
Hence, the number will be 234.

Test Level 3: Number System - 2 - Question 16

If the first 100 natural numbers are written side by side to make a big number, how many digits of this number will be even?

Detailed Solution for Test Level 3: Number System - 2 - Question 16

Single-digit even numbers are 4 in number (2, 4, 6 and 8).

In two-digit numbers, i.e. from 10 to 99,
0 comes at unit place in 10, 20, 30, 40 ....90 = 9 times
2 comes at unit place in 12, 22, 32, 42 ....92 = 9 times
2 comes at tens place in 20, 21, 22, 23, 24.....29 = 10 times
Thus, 2 comes from 10 to 99 = 9 + 10 = 19 times
Similarly 4, 6 and 8 come 19 times each.
In 100, there are 2 zeros.
Total number of even digits = 4 + 9 + 19 + 19 + 19 + 19 + 2 = 91

Test Level 3: Number System - 2 - Question 17

For how many integers n, with 10 ≤ n ≤ 100, is n2 + n - 90 divisible by 17?

Detailed Solution for Test Level 3: Number System - 2 - Question 17

Since n2 + n - 90 = (n - 9)(n + 10)
17 divides n2 + n - 90 if and only if 17 divides n - 9 or n + 10.
For n + 10 to be divisible by 17, n can be 24, 41, 58, 75 or 92.
For n - 9 to be divisible by 17, n can be 26, 43, 60, 77 or 94.
So, n has 10 values.

Test Level 3: Number System - 2 - Question 18

How many three-digit numbers do not change even if their digits are reversed?

Detailed Solution for Test Level 3: Number System - 2 - Question 18

A three-digit number doesn't change even if the digits are reversed when the first and last digits of the number are same.
Since the first digit cannot be zero, it can be anything from 1 to 9 and the second digit can be anything from 0 to 9.
So, the first and last digits can be filled in 9 ways and the middle can be filled in 10 ways.
So, total such numbers are 9 × 10 = 90.

Test Level 3: Number System - 2 - Question 19

Find the lower of the two successive natural numbers if the square of the sum of those numbers exceeds the sum of their squares by 112.

Detailed Solution for Test Level 3: Number System - 2 - Question 19

Again, spotting this with options is quite easy as we can see that 72 + 82 = 113 and that is 112 less than the value of (7 + 8)2 = 225.
Without options here you can think of a2 + b2 + 112 = (a + b)2 → 2ab = 112 or ab = 56.
Since, the numbers are consecutive, sifting through the factor pairs of 56 we can see the numbers as 7 & 8, respectively.

Test Level 3: Number System - 2 - Question 20

The product of a two-digit number by a number consisting of the same digits written in the reverse order is equal to 2430. Find the lower number.

Detailed Solution for Test Level 3: Number System - 2 - Question 20

A factor pair search of 2430 would give you the answer as 45 × 54.
Hence, 45 is the correct answer.

5 docs|272 tests
Information about Test Level 3: Number System - 2 Page
In this test you can find the Exam questions for Test Level 3: Number System - 2 solved & explained in the simplest way possible. Besides giving Questions and answers for Test Level 3: Number System - 2, EduRev gives you an ample number of Online tests for practice

Top Courses for CAT

Download as PDF

Top Courses for CAT