Number System Test

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Number System Test - 3

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Question 1
3 / -1

Raju had to divide 1080 by N, a two-digit number. Instead, he performed the division using M which is obtained by reversing the digits of N and ended up with a quotient which was 25 less than what he should have obtained otherwise. If 1080 is exactly divisible both by N and M, find the sum of the digits of N.

A
6

B
8

C
9

D
None of these

Solution

1080 = 2³ x 3² x 5¹.
Now, N x Q = 1080 and M x (Q - 25) = 1080
Do a bit of hit and trial.
N = 27
Question 2
3 / -1

A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is :

A
10

B
16

C
14

D
12

Solution

Let the rectangle has x and y tiles along its length and breadth respectively.
The number of white tiles
W = 2x + 2(y – 2) = 2 (x + y – 2)
And the number of red tiles = R = xy – 2 (x + y – 2)
Given that the number of white tiles is the same as the number of red tiles
⇒ 2 (x + y – 2) = xy – 2 (x + y – 2)
⇒ 4 (x + y – 2) = xy
⇒ xy – 4x – 4y = –8
⇒ (x – 4) (y – 4) = 8 = 8 ×1 or 4 × 2
⇒ m – 4 = 8 or 4
⇒m = 12 or 8
Therefore, the number of tiles along one edge of the floor can be 12
Question 3
3 / -1

A teacher wrote a number on the blackboard and the following observations were made by the students. The number is a four-digit number.The sum of the digits equals the product of the digits. The number is divisible by the sum of the digits.The sum of the digits of the number is

A
8

B
10

C
12

D
14

Solution

Using options, the only possible value is 4112. The key here is: The sum of the digits equals the product of the digits.
Question 4
3 / -1

The sum of the factorials of the three-digits of a 3-digit number is equal to the three-digit number formed by these three digits, taken in the same order. Which of the following is true of the number of such three-digit numbers, if no digit occurs more than once?

A
No such number exists

B
Exactly one such number exists

C
There is more than one such number, but they are finite in number

D
There are infinite such numbers

Solution

There is only one number, 145, which exhibits this property.
Question 5
3 / -1

Let S be a two-digit number such that both S and S² end with the same digit and none of the digits in S equals zero. When the digits of S are written in the reverse order, the square of the new number so obtained has the last digit as 6 and is less than 3000. How many values of S are possible?

A
3

B
4

C
5

D
6

Solution

The correct option is Option A.
In this question, several restrictions are operating: If S and S2 are ending with the same unit digit, then it can be 0, 1,5,6, but it is given that none of the digits is equal to zero, so the unit digit can be only 1, 5, 6. Next, the unit digit of the square of the number written in reverse order is 6, so the tens place digit of the actual number should be either 4 or 6.
So, the actual numbers could be 41, 45, 46, 61, 65, 66.
Now, this square is less than 3000, so the only possibilities are 41, 45, 46.
Question 6
3 / -1

Let N be a positive integer not equal to 1. Then none of the numbers 2, 3,...., N is a divisor of (N! - 1). Thus, we can conclude that

A
(N - 1) is a prime number

B
At least one of the numbers N + 1 N + 2,... ,N - 2 is a divisor of (N! - 1)

C
The smallest number between N and N! which is a divisor of (N! + 1), is a prime number

D
None of the foregoing statement is necessarily correct

Solution

Eliminate the options.
For example, option (1) can be eliminated by assuming N = 2
Question 7
3 / -1

16 students were writing a test in a class. Rahul made 14 mistakes in the paper, which was the highest number of mistakes made by any student. Which of the following statements is definitely true?

A
At least two students made the same number of mistakes

B
Exactly two students made the same number of mistakes

C
At most two students made the same number of mistakes

D
All students made different number of mistakes.

Solution

The number of mistakes made by all the students will be between 0 and 14, i.e., students are having a total of 15 options to make mistakes. Since the number of students = 16, at least two students will have the same number of mistakes (that can be zero also, i.e., two students are making no mistakes). Hence, option 1 is the answer.
Question 8
3 / -1

What is the least number of soldiers that can be drawn up in troops of 12, 15, 18 and 20 soldiers and also in form of a solid square?

A
900

B
400

C
1600

D
2500

Solution

In this type of question, We need to find out the LCM of the given numbers.
LCM of 12, 15, 18 and 20;
12 = 2 × 2 × 3;
15 = 3 × 5;
18 = 2 × 3 × 3;
20 = 2 × 2 × 5
Hence, LCM = 2 × 2 × 3 × 5 × 3 × 2 × 2 × 3 × 5 × 3
Since, the soldiers are in the form of a solid square.
Hence, LCM must be a perfect square. To make the LCM a perfect square, We have to multiply it by 5,
hence,
The required number of soldiers
= 2 × 2 × 3 × 3 × 5 × 5 × 2 × 2 × 3 × 3 × 5 × 5
= 900
Question 9
3 / -1

What is the remainder when (10³ + 9³)⁷⁵² is divided by 123?

A
10

B
729

C
752

D
1000

Solution

A remainder can never be greater than the no. which is the dividing factor.
hence, the remainder < 123
which leaves us with only one option, i.e. (a)
Question 10
3 / -1

When 7179 and 9699 are divided by another natural number N , remainder obtained is same. How many values of N will be ending with one or more than one zeroes?

A
24

B
124

C
18

D
None of these

Solution

9699 - 7129 = 2520.
Prime Factors of 2520: 1, 23, 32, 5, 7.
To get a zero at end: (5*2), (5*22), (5*23)-> 3 ways
& in the remaining numbers 1, 32, 7 -> possible combinations are 1,3,9,7, 21, 63 -> 6 ways.
possible ways of N => 6*3 ways= 18 ways
Question 11
3 / -1

Every element of S1 is made greater than or equal to every element of S2 by adding to each element of SI an integer x. Then, x cannot be less than

A
210

B
The smallest value of S2

C
The largest value of S2

D
(G-L)

Solution

For the least element L in S1 to be greater than the greatest element or equal to G in S2, the number that is added to L cannot be less than G-L.
Question 12
3 / -1

The History teacher was referring to a year in the 19th century. Rohan found an easy way to remember the year. He found that the number, when viewed in a mirror, increased 4.5 times. Which year was the teacher referring to?

A
1,801

B
1,810

C
1,818

D
More than one value

Solution

Going through the options, 8181/1818 = 4.5
Question 13
3 / -1

N is a number which when divided by 10 gives 9 as the remainder, when divided by 9 gives 8 as the remainder, when divided by 8 gives 7 as the remainder, when divided by 7 gives 6 as the remainder, when divided by 6 gives 5 as the remainder, when divided by 5 gives 4 as the remainder, when divided by 4 gives 3 as the remainder, when divided by 3 gives 2 as the remainder, when divided by 2 gives 1 as the remainder.What is N?

A
2519

B
841

C
839

D
2521

Solution

Check it through the options.
Alternatively, answer will be LCM (2,3,4,..., 9) - 1 = 2520 - 1 = 2519
Question 14
3 / -1

How many different four digit numbers are there in the octal (Base 8) system, expressed in that system?

A
3584

B
2058

C
6000

D
7000

Solution

The total number of numbers of four digits in octal system = 7 x 8 x 8 x 8 = 3584 When we convert this number into octal system, this is equal to 7000.
Question 15
3 / -1

When you reverse the digits of the number 13, the number increases by 18. How many other two digit numbers increase by 18 when their digits are reversed?

A
5

B
6

C
7

D
8
Question 16
3 / -1

Find the unit digit:
346 ⁷⁶⁵ * 768 ⁹⁸³ * 987 ⁵⁹⁹

A
3

B
5

C
6

D
9

Solution

In this type of problem
Step 1: we find the unit digit of each term
Step 2: we find the product of the unit digits of each term
Step 3: The unit digit of the product will be the product of whole number
The unit digit of 346 ⁷⁶⁵ = 6
The unit digit of 768 ⁹⁸³= 2
The unit digit of 987⁵⁹⁹= 3
6 * 2 * 3 = 36
Hence, the unit digit is 6.
Question 17
3 / -1

If a number when successively divided by 9, 7, and 5 leaves the remainders 8, 5, and 1 respectively. Find the smallest such 4-digit number.

A
1660

B
1260

C
2016

D
1061

Solution

This is a problem of successive division.
Divisors 9 7 5
Remainders 8 5 1
The smallest such numbers will be {(1 * 7) + 5} * 9 + 8} Which will be equal to 116.
Now, the general form of the numbers will be:
(9 * 7 * 5) k + 116
= 315k + 116
When we put k = 3
Then the number we obtain is 315 * 3 + 116
The largest such 3-digit number is 1061.
Question 18
3 / -1

A number when divided by 841 gives a remainder of 87. What will be the remainder when we divide the same number by 29?

A
3

B
5

C
2

D
0

Solution

Let the number be N and its quotient be k.
Then the number N can be written in the form of:
N = 841k + 87
Now, we have to find out the what will be the remainder when it is divided by 29.
The number is (841k + 87)
Let’s divide it by 29
(841k + 87)/ 29
841 and 87 both are completely divisible by 29.
Therefore, the remainder when the number N is divided by 29 is 0.
Question 19
3 / -1

A number when divided by 703 gives a remainder of 75. What will be the remainder when we divide the same number by 37?

A
1

B
2

C
5

D
7

Solution

Let the number be N and its quotient be k.
Then the number N can be written in the form of:
N = 703k + 75
Now, we have to find out the what will be the remainder when it is divided by 37.
The number is (703k + 75)
Let’s divide it by 37
(703k + 75)/ 37
703 is divisible by 37 hence, remainder will be 0 whereas, 75 when divided by 37 leaves remainder 1.
Therefore, the remainder when the number N is divided by 37 will be (0 + 1) i.e. 1.
Question 20
3 / -1

Find the remainder when a number is divided by 49. The same number when divided by 147 leaves 53 as remainder.

A
1

B
2

C
3

D
4

Solution

Let the number be N and its quotient be k.
Then the number N can be written in the form of:
N = 147k + 53
Now, we have to find out the what will be the remainder when it is divided by 49.
The number is (147k + 53)
Let’s divide it by 49
(147k + 53)/ 49
147 is divisible by 49 hence, remainder will be 0 whereas, 53 when divided by 49 leaves 1 as remainder.
Therefore, the remainder when the number N is divided by 49 will be (0 + 4) i.e. 4.

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Number System Test - 3

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Number System Test - 3

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Question 1

6

8

9

None of these

Solution

Question 2

10

16

14

12

Solution

Question 3

8

10

12

14

Solution

Question 4

No such number exists

Exactly one such number exists

There is more than one such number, but they are finite in number

There are infinite such numbers

Solution

Question 5

3

4

5

6

Solution

Question 6

(N - 1) is a prime number

At least one of the numbers N + 1 N + 2,... ,N - 2 is a divisor of (N! - 1)

The smallest number between N and N! which is a divisor of (N! + 1), is a prime number

None of the foregoing statement is necessarily correct

Solution

Question 7

At least two students made the same number of mistakes

Exactly two students made the same number of mistakes

At most two students made the same number of mistakes

All students made different number of mistakes.

Solution

Question 8

900

400

1600

2500

Solution

Question 9

10

729

752

1000

Solution

Question 10

24

124

18

None of these

Solution

Question 11

210

The smallest value of S2

The largest value of S2

(G-L)

Solution

Question 12

1,801

1,810

1,818

More than one value

Solution