Problem Solving

Number systems sample questions with answers:
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1>If both 11^2 and 3^3 are factors of the number a * 4^3 * 6^2 * 13^11, then what is the smallest possible value of a?


(1)121
(2)3267
(3)363
(4)33
Correct choice (3). Correct Answer - (363)


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Explanatory Answers

11^2 is a factor of the given number. The number does not have a power or multiple of 11 as its factor. Hence, "a" should include 11^2

3^3 is a factor of the given number. 6^2 is a part of the number. 6^2 has 3^2 in it. Therefore, if 3^3 has to be a factor of the given number a * 4^3 * 6^2 * 13^11, then we will need at least another 3.

Therefore, if "a" should be at least 11^2 * 3 = 363 if the given number has to have 11^2 and 3^3 as its factors.


2>Find the greatest number of five digits, which is exactly divisible by 7, 10, 15, 21 and 28.

(1) 99840
(2) 99900
(3) 99960
(4) 99990

Correct Choice is (3) and the correct answer is 99960


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Explanatory Answers

The number should be exactly divisible by 15 (3, 5), 21 (3, 7), 28 (4, 7).
Hence, it is enough to check the divisibility for 3, 4, 5 and 7.

99960 is the only number which satisfies the given condition.



3>The largest number amongst the following that will perfectly divide 101^100 - 1 is

(1) 100
(2) 10,000
(3) 100100
(4) 100,000

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



Explanatory Answer
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101^2 = 10201.
101^2 - 1 = 10200. This is divisible by 100.

Similarly try for 101^3 - 1 = 1030301 - 1 = 1030300.

So you can safely conclude that (101^1 - 1) to (101^9 - 1) will be divisible by 100.
(101^10 - 1) to (101^99 - 1) will be divisible by 1000.
Therefore, (101^100 - 1) will be divisible by 10,000.


5>Let x, y and z be distinct integers. x and y are odd and positive, and z is even and positive. Which one of the following statements cannot be true?

(1) (x-z)^2y is even
(2) (x-z)y^2 is odd
(3) (x-z)y is odd
(4) (x-y)^2z is even

Correct Choice is (1) and Correct Answer is (x-z)^2 y is even


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Explanatory Answer

x and y are odd and positive and z is even and positive

(x - z)^2y is even cannot be true

x - z is odd and y is odd
Therefore, (x - z)^2 will be odd and (x - z)^2 y will be odd






6>When a number is divided by 36, it leaves a remainder of 19. What will be the remainder when the number is divided by 12?

(1) 10
(2) 7
(3) 192
(4) None of these

Correct Choice is (2) and Correct Answer is 7


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Explanatory Answer

Let the number be 'a'.

When 'a' is divided by 36, let the quotient be 'q' and we know the remainder is 19

i.e., and remainder is 19

or a = 36q + 19

when a is divided by 12, we get

or

36q is perfectly divided by 12

Therefore, remainder = 7




7>The sum of the first 100 numbers, 1 to 100 is divisible by

(1) 2, 4 and 8
(2) 2 and 4
(3) 2 only
(4) None of these

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


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Explanatory Answer

The sum of the first 100 natural numbers is given by (n(n + 1))/2 = (100(101))/2 = 50(101).

101 is an odd number and 50 is divisible by 2. Hence, 50*101 will be divisible by 2.




8>How many different factors are there for the number 48, excluding 1 and 48?

(1) 12
(2) 4
(3) 8
(4) None of these

Correct Choice is (3) and the correct answer is 8


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Explanatory Answers

To find the number of factors of a given number, express the number as a product of powers of prime numbers.

In this case, 48 can be written as 16 * 3 = (2^4 * 3)

Now, increment the power of each of the prime numbers by 1 and multiply the result.

In this case it will be (4 + 1)*(1 + 1) = 5 * 2 = 10 (the power of 2 is 4 and the power of 3 is 1)

Therefore, there will 10 factors including 1 and 48. Excluding, these two numbers, you will have 10 - 2 = 8 factors.



9>10^25 - 7 is divisible by


(1) 2
(2) 3
(3) 9
(4)Both (2) and (3)
Correct Answer - 3. Correct choice is (2)


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Explanatory Answers

10^25 - 7 = (10^25 - 1) - 6

The number 10^25 - 1 = 99.....9 (25 digits) is divisible by 3 and 9.

Therefore, (10^25 - 1) - 6 = (24 nines and unit digit is 3) 99.......93.

This number is only divisible by 3 (from the given choices).




10>Find the G.C.D of 12x^2y^3z^2, 18x^3y^2z^4, and 24xy^4z^3

(1) 6xy^2z^2
(2) 6x^3y^4z^3
(3) 24xy^2z^2
(4) 18x^2y^2z^3

Correct Choice is (1) and Correct Answer is 6xy^2z^2


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Explanatory Answer

G.C.D of 12, 18 and 24 is 6.

The common factors are x, y, z and their highest powers common to all are 1, 2 and 2 respectively.

Therefore, G.C.D = 6xy^2z^2


11>What is the value of M and N respectively? If M39048458N is divisible by 8 and 11; Where M and N are single digit integers?

(1) 7, 8
(2) 8, 6
(3) 6, 4
(4) 5, 4

Correct Choice is (3) and correct answer is 6, 4


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Explanatory Answers

If the last three digits of a number is divisible by 8, then the number is divisible by 8 (test of divisibility by 8).

Here, last three digits 58N is divisible by 8 if N = 4. (Since 584 is divisible by 8.)

For divisibility by 11. If the sum of digits at odd and even places of a given number are equal or differ by a number divisible by 11, then the given number is divisible by 11.

Therefore, (M+9+4+4+8)-(3+0+8+5+N)=(M+5) {putting N=4}should be divisible by 11 => when M = 6.





12>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 =(578*374)/(34*34) = 187 marbles.


13>What number should be subtracted from x^3 + 4x^2 - 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 ,f(x)/x+a then the remainder is f(-a).

In this case, as x + 3 divides x^3 + 4x^2 - 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




14>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






15>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 17/d leaves a remainder of 4.

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






16>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.




17>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.



18>A person starts multiplying consecutive positive integers from 20. How many numbers should he multiply before he 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.




19>For what value of 'n' will the remainder of 351^n and 352^n 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 351^2 is divided by 7, the remainder will be 1^2 = 1.

When 352^2 is divided by 7, the remainder will be 2^2 = 4.

So when n = 2, the remainders are different.

When n = 3,

When 351^3 is divided by 7, the remainder will be 1^3 = 1.

When 352^3 is divided by 7, the remainder will be 2^3 = 8.

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

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





20>What is the remainder when 9^1 + 9^2 + 9^3 + .... + 9^8 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 9^1 + 9^2 + 9^3 + ..... + 9^8 is divided by 6 will be equal to '0'.

if 11^2 and 3^3 are the factors of the no. , then they should completely divide the no.
putting value 33,121,363..if u put a =363, no gets completely divide. Hence it is the solution

we require 11*11* 3 in the deno..

why ?


cancel the common factores in numero and denomo...

IMO 3