Number Theory : Test of Divisibility Question
  

Question

The largest number amongst the following that will perfectly divide 101¹⁰⁰ - 1 is

1. 100
2. 10,000
3. 100¹⁰⁰
4. 100,000

Correct Choice is (2) and Correct Answer is 10,000

Explanatory Answer

The easiest way to solve such problems for CAT puposes is trial and error or by back substituting answers in the choices given.

101² = 10201.
101² - 1 = 10200. This is divisible by 100.

Similarly try for 101³ - 1 = 1030301 - 1 = 1030300.

So you can safely conclude that (101¹ - 1) to (101⁹ - 1) will be divisible by 100.
(101¹⁰ - 1) to (101⁹⁹ - 1) will be divisible by 1000.
Therefore, (101¹⁰⁰ - 1) will be divisible by 10,000. 

Question

A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?

1. 13
2. 59
3. 35
4. 37

Correct Choice - (4). Correct Answer is 37

Explanatory Answer

Let the original number be 'a'
Let the divisor be 'd'

Let the quotient of the division of a by d be 'x'

Therefore, we can write the relation as  = x and the remainder is 24.
i.e., a = dx + 24

When twice the original number is divided by d, 2a is divided by d.
We know that a = dx + 24. Therefore, 2a = 2dx + 48

The problem states that  leaves a remainder of 11.
2dx is perfectly divisible by d and will therefore, not leave a remainder.

The remainder of 11 was obtained by dividing 48 by d.
When 48 is divided by 37, the remainder that one will obtain is 11.
Hence, the divisor is 37.

Question

How many keystrokes are needed to type numbers from 1 to 1000?

1. 3001
2. 2893
3. 2704
4. 2890

Correct Choice is (2) and Correct Answer is 2893

Explanatory Answer

While typing numbers from 1 to 1000, you have 9 single digit numbers from 1 to 9. Each of them require one keystroke. That is 9 key strokes.

There are 90 two-digit numbers, from 10 to 99. Each of these numbers require 2 keystrokes. Therefore, one requires 180 keystrokes to type the 2 digit numbers.

There are 900 three-digit numbers, from 100 to 999. Each of these numbers require 3 keystrokes. Therefore, one requires 2700 keystrokes to type these 3 digit numbers.

Then 1000 is a four-digit number which requires 4 keystrokes.

Totally, therefore, one requires 9 + 180 + 2700 + 4 = 2893 keystrokes.

Question

When 242 is divided by a certain divisor the remainder obtained is 8. When 698 is divided by the same divisor the remainder obtained is 9. However, when the sum of the two numbers 242 and 698 is divided by the divisor, the remainder obtained is 4. What is the value of the divisor?

1. 11
2. 17
3. 13
4. 23

Correct Choice is (3) and Correct Answer is 13

Explanatory Answer

Let the divisor be d.

When 242 is divided by the divisor, let the quotient be 'x' and we know that the remainder is 8.
Therefore, 242 = xd + 8

Similarly, let y be the quotient when 698 is divided by d.
Then, 698 = yd + 9.

242 + 698 = 940 = xd + yd + 8 + 9
940 = xd + yd + 17
As xd and yd are divisible by d, the remainder when 940 is divided by d should have been 17.

However, as the question states that the remainder is 4, it would be possible only when leaves a remainder of 4.

If the remainder obtained is 4 when 17 is divided by d, then d has to be 13.

Question

Let n be the number of different 5 digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5 and 6, no digit being repeated in the numbers. What is the value of n?

1. 144
2. 168
3. 192
4. None of these

Correct Choice is (3) and correct answer is 192

Explanatory Answer

Test of divisibility by 4 is that the last two digits should be divisible by 4.

Case 1 : When the last 2 digits are 12, i.e., _ _ _ 12 = 4 * 3 * 2 = 24 numbers

Case 2 : When the last 2 digits are 16, there are 24 numbers

Case 3 : When the last 2 digits are 24 there are 24 numbers

Case 4 : When the last 2 digits are 32 there are 24numbers

Case 5 : When last 2 digits are 36 there are 24 numbers

Case 6 : When last 2 digits are 52 there are 24 numbers

Case 7 : When last 2 digits are 56 there are 24 numbers

Case 8 : When last 2 digits are 64 there are 24 numbers

Total = 8 * 24 = 192

Question

What number should be subtracted from x³ + 4x² - 7x + 12 if it is to be perfectly divisible by x + 3?

1. 42
2. 39
3. 13
4. None of these

Correct Choice is (1) and Correct Answer is 42

Explanatory Answer

According to remainder theorem when , then the remainder is f(-a).

In this case, as x + 3 divides x³ + 4x² - 7x + 12 - k perfectly (k being the number to be subtracted), the remainder is 0 when the value of x is substituted by -3.

i.e., (-3)³ + 4(-3)² - 7(-3) + 12 - k = 0

or -27 + 36 + 21 + 12 = k

or k = 42

Question

What is the minimum number of square marbles required to tile a floor of length 5 metres 78 cm and width 3 metres 74 cm?

1. 176
2. 187
3. 54043
4. 748

Correct Choice is (2) and correct answer is 187

Explanatory Answer

The marbles used to tile the floor are square marbles. Therefore, the length of the marble = width of the marble.
As we have to use whole number of marbles, the side of the square should a factor of both 5 m 78 cm and 3m 74. And it should be the highest factor of 5 m 78 cm and 3m 74.

5 m 78 cm = 578 cm and 3 m 74 cm = 374 cm.
The HCF of 578 and 374 = 34.

Hence, the side of the square is 34.

The number of such square marbles required =  = 187 marbles.

Question

A person starts multiplying consecutive positive integers from 20. How many numbers should he multiply before the will have result that will end with 3 zeroes?

1. 11
2. 10
3. 6
4. 5

Correct Choice is (3) and correct answer is 6

Explanatory Answer

A number will end in 3 zeroes when it is multiplied by 3 10s.
To get a 10, one needs a 5 and a 2.

Therefore, this person should multiply till he encounters three 5s and three 2s.
20 has one 5 (5 * 4) and 25 has two 5s (5 * 5).
20 has two 2s (5 * 2 * 2) and 22 has one 2 (11 * 2).

Therefore, he has to multiply till 25 to get three 5s and three 2s, that will make three 10s.
So, he has to multiply from 20 to 25 i.e. 6 numbers.

Question

For what value of 'n' will the remainder of 351ⁿ and 352ⁿ be the same when divided by 7?

1. 2
2. 3
3. 6
4. 4

Correct Choice is (2) and the Correct Answer is 3

Explanatory Answer

When 351 is divided by 7, the remainder is 1.

When 352 is divided by 7, the remainder is 2.

Let us look at answer choice (1), n = 2

When 351² is divided by 7, the remainder will be 1² = 1.

When 352² is divided by 7, the remainder will be 2² = 4.

So when n = 2, the remainders are different.

When n = 3,

When 351³ is divided by 7, the remainder will be 1³ = 1.

When 352³ is divided by 7, the remainder will be 2³ = 8.

As 8 is greater than 7, divide 8 again by 7, the new remainder is 1.

So when n = 3, both 351ⁿ and 352ⁿ will have the same remainder when divided by 7.

Question

What is the remainder when 9¹ + 9² + 9³ + .... + 9⁸ is divided by 6?>

1. 3
2. 2
3. 0
4. 5

Correct Choice is (3) and Correct Answer is 0

Explanatory Answer

6 is an even multiple of 3. When any even multiple of 3 is divided by 6, it will leave a remainder of 0. Or in other words it is perfectly divisible by 6.

On the contrary, when any odd multiple of 3 is divided by 6, it will leave a remainder of 3. For e.g when 9 an odd multiple of 3 is divided by 6, you will get a remainder of 3.

9 is an odd multiple of 3. And all powers of 9 are odd multiples of 3.
Therefore, when each of the 8 powers of 9 listed above are divided by 6, each of them will leave a remainder of 3.

The total value of the remainder = 3 + 3 + .... + 3 (8 remainders) = 24.
24 is divisible by 6. Hence, it will leave no remainder.

Hence, the final remainder when the expression 9¹ + 9² + 9³ + .... + 9⁸ is divided by 6 will be equal to '0'.

Question

Ram and Shyam take a vacation at their grandparents' house. During the vacation, they do any activity together. They either played tennis in the evening or practiced Yoga in the morning, ensuring that they do not undertake both the activities on any single day. There were some days when they did nothing. Out of the days that they stayed at their grandparents' house, they involved in one of the two activities on 22 days. However, their grandmother while sending an end of vacation report to their parents stated that they did not do anything on 24 mornings and they did nothing on 12 evenings. How long was their vacation?

1. 36 days
2. 14 days
3. 19 days
4. Cannot be determined
   

Correct Choice is (3) and Correct Answer is 19 days

Explanatory Answer

Let the number of days that they holidayed be equal to T.

Then, they practiced Yoga on (T – 24) mornings.
They played tennis on (T – 12) evenings.

As they did not do both the activities together on any single day,

Days on which they had any activity = Number of days they practiced Yoga + Number of days they played tennis

i.e., 22 = T – 24 + T – 12

or 22 + 24 +12 = 2T

or 58 = 2T

Hence, T = 29.

Question

N is the smallest number that has 5 factors. How many factors does (N - 1) have??

1. 2
2. 3
3. 4
4. 5

Correct Choice is (c) and the Correct Answer is 4

Explanatory Answer

A number that has 5 factors has to be of the form p⁴ where 'p' is a prime number.

The smallest such number is 2⁴ = 16

Therefore, N - 1 = 15.

The factors of 15 are 1, 3, 5, 15.

So, N - 1 has 4 factors.