- Total number of questions : 20
- Time allotted : 30:00 Minutes:Seconds.
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1.

A set of S consists of,

i). All odd numbers from 1 to 55.

ii). All even numbers from 56 to 150.

What is the index of the highest power of 3 in the product of all the elements of the set SS?

2.

A and B are playing mathematical puzzles. A asks B "the whole numbers, greater than one, which can divide all the nine three digit numbers 111,222,333,444,555,666, 777,888 and 999?"

B immediately gave the desired answer. It was:

3.

The remainder when the positive integer m is divided by 7 is x. The remainder when m is divided by 14 is x+7.

Which one of the following could m equal?

4.

The positive integers mm and nn leave remainders of 2 and 3, respectively, when divided by 6. m>nm>n.

What is the remainder when m–nm–n is divided by 6?

5.

A chain smoker had spent all the money he had. He had no money to buy his cigarettes. Hence, he resorted to join the stubs and to smoke them.

He needed 4 stubs to make a single cigarette. If he got a pack of 10 cigarettes as a gift, then how many cigarettes could he smoke in all?

6.

When writing numbers from 1 to 10,000, how many times is the digit 9 written ?

7.

What is the remainder when $3^7$ is divided by $8$?

8.

\(\left ( 64 - 12 \right )^{2} + 4 × 64 × 12 = ?\)

9.

The smallest prime number is?

10.

The unit digit in \(7^{153}\) is?

11.

The largest 4 digit number exactly divisible by 88 is:

12.

How many of the following numbers are divisible by 132 ?

264, 396, 462, 792, 968, 2178, 5184, 6336

13.

\((1000)^{9} \div 10^{24} = ?\)

14.

\(\left \{(476 + 424)^{2} - 4 \times 476 \times 424 \right \} = ?\)

15.

A number when divided by 6 leaves a remainder 3. When the square of the same number is divided by 6, the remainder is:

16.

The sum of the present ages of two persons A and B is 60. If the age of A is twice that of B, find the sum of their ages 5 years hence?

17.

The sum of three prime numbers is 100. If one of them exceeds another by 36, then one of the numbers is:

18.

What least number must be added to 1056, so that the sum is completely divisible by 23 ?

19.

It is being given that \((2^{32} + 1)\) is completely divisible by a whole number. Which of the following numbers is completely divisible by this number?

20.

Which one of the following is not a prime number?

Total number of questions : 20

Number of answered questions : 0

Number of unanswered questions : 20

1.

A set of S consists of,

i). All odd numbers from 1 to 55.

ii). All even numbers from 56 to 150.

What is the index of the highest power of 3 in the product of all the elements of the set SS?

Your Answer : (Not Answered)

Correct Answer : **A**

No answer description available for this question.

2.

A and B are playing mathematical puzzles. A asks B "the whole numbers, greater than one, which can divide all the nine three digit numbers 111,222,333,444,555,666, 777,888 and 999?"

B immediately gave the desired answer. It was:

Your Answer : (Not Answered)

Correct Answer : **B**

Each of the number can be written as a multiple of 111.

The factors of 111 are 3 and 37

Thus the desired answer is **3, 37 and 111**

3.

The remainder when the positive integer m is divided by 7 is x. The remainder when m is divided by 14 is x+7.

Which one of the following could m equal?

Your Answer : (Not Answered)

Correct Answer : **B**

No answer description available for this question.

4.

The positive integers mm and nn leave remainders of 2 and 3, respectively, when divided by 6. m>nm>n.

What is the remainder when m–nm–n is divided by 6?

Your Answer : (Not Answered)

Correct Answer : **C**

We are given that the numbers mm and nn, when divided by 6, leave remainders of 2 and 3, respectively.

Hence, we can represent the numbers mm and nn as 6p+26p+2 and 6q+36q+3, respectively, where pp and qq are suitable integers.

Now,

m−n=(6p+2)−(6q+3)

=6p−6q−1

=6(p−q)−1

A remainder must be positive, so let’s add 6 to this expression and compensate by subtracting 6:

6(p−q)−1=6(p−q)−6+6−1

=6(p−q)−6+5

=6(p−q−1)+5

Thus, the remainder is **5**

5.

A chain smoker had spent all the money he had. He had no money to buy his cigarettes. Hence, he resorted to join the stubs and to smoke them.

He needed 4 stubs to make a single cigarette. If he got a pack of 10 cigarettes as a gift, then how many cigarettes could he smoke in all?

Your Answer : (Not Answered)

Correct Answer : **D**

Ten cigarettes give 10 stub.

From 10 stubs 3 more cigarettes can be made (2 stubs would be obtained from 2 cigarettes formed by joining 8 stubs.)

So he could smoke **13 cigarettes** in all.

6.

When writing numbers from 1 to 10,000, how many times is the digit 9 written ?

Your Answer : (Not Answered)

Correct Answer : **C**

The digits 9 occurs in the thousands place in 1000 numbers.

It occurs in the hundreds place in 1000 numbers and so on

The digit occurs **4000 times**.

7.

What is the remainder when $3^7$ is divided by $8$?

Your Answer : (Not Answered)

Correct Answer : **C**

No answer description available for this question.

8.

\(\left ( 64 - 12 \right )^{2} + 4 × 64 × 12 = ?\)

Your Answer : (Not Answered)

Correct Answer : **D**

Given statement is like \(\left ( a - b \right )^{2} + 4ab\) where \(a=64\) and \(b=12\)

\(\left ( a - b \right )^{2} + 4ab\)

\(=\left (a^{2} - 2ab+b^{2}\right )+4ab\)

\(=a^{2}+2ab+b^{2}\)

\(=\left ( a + b \right )^{2}\)

Hence,

\(\left ( 64 - 12 \right )^{2} + 4 × 64 × 12\)

\(=\left ( 64 + 12 \right )^{2}\)

\(=76^{2}\)

\(=5776\)

9.

The smallest prime number is?

Your Answer : (Not Answered)

Correct Answer : **C**

The smallest prime number is 2

10.

The unit digit in \(7^{153}\) is?

Your Answer : (Not Answered)

Correct Answer : **C**

\(7^{153} = \left ( 7^{4}\right )^{38}\times 7\)

Now, unit digit of \(\left ( 7^{4}\right )^{38} = 1\)

Therefore, unit digit of \(7153 = 1 \times 7\)

\(= 7\)

11.

The largest 4 digit number exactly divisible by 88 is:

Your Answer : (Not Answered)

Correct Answer : **A**

Largest 4-digit number = 9999

\(\frac{9999}{88} = 113\tfrac{55}{88}\)

Required number \(= (9999 - 55) = 9944\)

12.

How many of the following numbers are divisible by 132 ?

264, 396, 462, 792, 968, 2178, 5184, 6336

Your Answer : (Not Answered)

Correct Answer : **A**

\(132 = 4 \times 3 \times 11\)

So, if the number divisible by all the three number 4, 3 and 11, then the number is divisible by 132 also.

\(264 \Rightarrow 11,3,4\)

\(396 \Rightarrow 11,3,4\)

\(462 \Rightarrow 11,3\)

\(792 \Rightarrow 11,3,4\)

\(968 \Rightarrow 11,4\)

\(2178 \Rightarrow 11,3\)

\(5184 \Rightarrow 3,4\)

\(6336 \Rightarrow 11,3,4\)

Therefore the following numbers are divisible by 132 : 264, 396, 792 and 6336.

Required number of number = 4.

13.

\((1000)^{9} \div 10^{24} = ?\)

Your Answer : (Not Answered)

Correct Answer : **B**

\(=\dfrac{(1000)^{9}}{(10)^{24}}\\ =\dfrac{(10^{3})^{9}}{(10)^{24}}\\ =\dfrac{(10)^{27}}{(10)^{24}}\\ =10^{(27-24)}\\ =10^{3}\\ = 1000\)

14.

\(\left \{(476 + 424)^{2} - 4 \times 476 \times 424 \right \} = ?\)

Your Answer : (Not Answered)

Correct Answer : **C**

Given Expression = \([(a + b)^{2} - 4ab]\), where \(a = 476\) and \(b = 424\)

\(=[(476 + 424)^{2} - 4 \times 476 \times 424]\)

\(= [(900)^{2} - 807296]\)

\(= 810000 - 807296\)

\(= 2704\)

15.

Your Answer : (Not Answered)

Correct Answer : **D**

Let \(x=6q+3\). Then,

\(x^{2} = (6q+3)^{2} \\= 36q^{2}+36q+9 \\= 6(6q^{2}+6q+1)+3\)

So, when \(2n\) is divided by \(4\), remainder \(=3\)

16.

Your Answer : (Not Answered)

Correct Answer : **C**

\(A + B = 60, A = 2B\)

\(2B + B = 60 => B = 20\) then \(A = 40\)

5 years, their ages will be 45 and 25

Sum of their ages \(= 45 + 25 = 70\)

17.

Your Answer : (Not Answered)

Correct Answer : **D**

\(x+(x+36)+y=100 \\ \Rightarrow 2x+y=64\)

Therefore \(y\) must be even prime, which is 2.

Therefore,

\(2x+2=64 \\ \Rightarrow x=31\)

Third prime number

\(= (x+36)\\= (31+36)\\= 67\)

18.

What least number must be added to 1056, so that the sum is completely divisible by 23 ?

Your Answer : (Not Answered)

Correct Answer : **A**

\(\dfrac{1056}{23}=45.91\\ 23 \times 45=1035\\ 1056-1035=21\\ \therefore 23-21=2\)

19.

Your Answer : (Not Answered)

Correct Answer : **D**

Let \(2^{32} = x\).

Then, \((2^{32} + 1) = (x + 1)\).

Let \((x + 1)\) be completely divisible by the natural number N. Then,

\((2^{96} + 1) \\= [(2^{32})^{3} + 1]\\= (x^{3} + 1)\\= (x + 1)(x^{2} - x + 1)\)

which is completely divisible by N, since \((x + 1)\) is divisible by N.

20.

Which one of the following is not a prime number?

Your Answer : (Not Answered)

Correct Answer : **D**

91 is divisible by 7. So, it is not a prime number.

Total number of questions : 20

Number of answered questions : 0

Number of unanswered questions : 20

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