TYPES OF NUMBERS

• Integers : All numbers whose fractional part is 0 (zero) like -3, -2, 1, 0, 10, 100 are integers.
• Natural Numbers : Counting numbers like 1, 2, 3, 4, 5, 6 … Basically, all integers greater than 0 are natural numbers … More on Numbers
 Question 1
Which is not the prime number?
 A 43 B 57 C 73 D 101
Arithmetic Aptitude    Numbers
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Question 1 Explanation:
A positive natural number is called prime number if nothing divides it except the number itself and 1. 57 is not a prime number as it is divisible by 3 and 19.
 Question 2
How many terms are there in 3,9,27,81........531441?
 A 25 B 12 C 13 D 14
Arithmetic Aptitude 4    Numbers
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Question 2 Explanation:
```3, 9, 27, 81..............531441 form a G.P.
with a = 3 and r = 9/3 = 3

Let the number of terms be n

Then 3 x 3n-1 = 531441

∴ 3n = 312

∴ n = 12```
 Question 3
If the average of four consecutive odd numbers is 12, find the smallest of these numbers?
 A 5 B 7 C 9 D 11
Arithmetic Aptitude 4    Numbers
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Question 3 Explanation:
```Let the numbers be x, x+2, x+4 and x+6
Then (x + x + 2 + x + 4 + x + 6)/4  = 12
∴ 4x + 12 = 48
∴ x = 9 ```
 Question 4
If the sum of two numbers is 13 and the sum of their square is 85. Find the numbers?
 A 6 & 7 B 5 & 8 C 4 & 9 D 3 & 10
Arithmetic Aptitude 4    Numbers
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Question 4 Explanation:
Let the numbers be x and 13-x Then x2 + (13 – x)2 = 85 ∴ x2 + 169 + x2 – 26x = 85 ∴ 2 x2 – 26x + 84 = 0 ∴ x2 – 13x + 42 = 0 ∴ (x-6)(x-7)=0 Hence numbers are 6 & 7
 Question 5
The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 45. What is the difference between the two digits of that number?
 A 5 B 7 C 6 D None of these
Arithmetic Aptitude 5    Numbers
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Question 5 Explanation:
```Let the ten’s digit be x and unit’s digit be y
Then (10x + y) – (10y + x) = 45
9(x – y) = 45
x – y = 5```
 Question 6
A two-digit number is such that the product of the digits is 12. When 9 is subtracted from the number, the digits are reversed. The number is:
 A 34 B 62 C 43 D 26
Arithmetic Aptitude 5    Numbers
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Question 6 Explanation:
```Let the ten’s and unit’s digit be x and y.

Then 10x + - 9 = 10 x + x

10x2 + 12 -9x = 120 + x2
9x2 – 9x – 108 = 0
x2 –x – 12 = 0
x2 –4x + 3x – 12 = 0
(x – 4) (x + 3) = 0
Therefore x = 4
Hence the required no. is 43```
 Question 7
Find a positive number which when increased by 16 is equal to 80 times the reciprocal of the number
 A 20 B -4 C -10 D 4
Arithmetic Aptitude 5    Numbers
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Question 7 Explanation:
```Let the number be x.

Then x + 16 = 80 * (1/x)

x2 + 16x – 80 = 0
x2 + 20x – 4x – 80 =0
(x + 20) (x -4)
Therefore x = 4```
 Question 8
What is the sum of two consecutive odd numbers, the difference of whose squares is 56?
 A 30 B 28 C 34 D 32
Arithmetic Aptitude 5    Numbers
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Question 8 Explanation:
```Let the no. be x and (x +2).
Then (x +2)2 – x2 = 56
4x + 4 = 56
x + 1 = 14
x = 13
Sum of numbers = x + (x +2) = 28```
 Question 9
The product of two numbers is 108 and the sum of their squares is 225. The difference of the number is:
 A 5 B 4 C 3 D None of these
Arithmetic Aptitude 5    Numbers
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Question 9 Explanation:
```Let the numbers be x and y.
Then xy = 108 and x2 + y2 = 225
(x –y)2 = x2 + y2 – 2xy
(x –y)2 = 225 – 216
(x –y)2 = 9
Therefore (x –y) = 3```
 Question 10
The average of 21 results is 20. Average of 1st 10 of them is 24 that of last 10 is 14. the result of 11'th is :
 A 42 B 44 C 46 D 40
Arithmetic Aptitude 6    Numbers
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Question 10 Explanation:
``` 11'th result = sum of 21 results – sum of 20 results
= 21 x 20 – (24 x 10 + 14 x 10)
= 420 – (240 + 140)
= 420- 380 = 40```
 Question 11
What could be the maximum value of Y in the following equation given that neither of X, Y, Z is zero? 5X8 + 3Y4 + 2Z1 = 1103
 A 0 B 7 C 8 D 9
Numbers    Simplification and Approximation
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Question 11 Explanation:
```  1 1    <- CARRY

5 X 8
+ 3 Y 4
+ 2 Z 1
--------
11 0 3
--------
```
Clearly, X + Y + Z + 1 = 10 => X + Y + Z = 9 Now, since neither of X, Y, Z can be zero, the value of Y will be maximum when X = Z = 1. => Max Y = 7
 Question 12
Which of the following are prime numbers ? (i) 147 (ii) 327 (iii) 547 (iv) 637
 A 147 B 327 C 547 D 637
Numbers
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Question 12 Explanation:
(i) 147 132 = 169 > 147. Prime numbers less than 13 are 2, 3, 5, 7, 11. 147 is divisible by 3. Therefore, 147 is not a prime number. It is a composite number.   (ii) 327 192 = 361 > 327. Prime numbers less than 19 are 2, 3, 5, 7, 11, 13, 17. 327 is divisible by 3. Therefore, 327 is not a prime number. It is a composite number.   (iii) 547 242 = 576 > 547. Prime numbers less than 24 are 2, 3, 5, 7, 11, 13, 17, 19, 23. 547 is not divisible by any of the above prime numbers. Therefore, it is a prime number.   (iv) 637 262 = 676 > 637. Prime numbers less than 26 are 2, 3, 5, 7, 11, 13, 17, 19, 23. 637 is divisible by 7. Therefore, 637 is not a prime number. It is a composite number.   Hence, 547 is the only prime number among the given numbers.
 Question 13
What is the unit's digit in the product (267)153 x (66666)72 ?
 A 7 B 6 C 1 D 2
Numbers    Simplification and Approximation
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Question 13 Explanation:
The unit's digit in (267)153 x (66666)72 is same as unit's digit in (7)153 x (6)72. Now, unit's digit in 74 is 1. => Unit's digit in 7z is 1, where 'z' is a multiple of 4. 152 is the nearest number to 153, which is a multiple of 4. => Unit's digit in 7152 is 1 => Unit's digit in 7153 is 1 x 7 = 7   Also, unit's digit when 6 is raised to any power remains 6, i.e., for all 'n', 6n will have 6 in the unit's place. Therefore, unit's digit in (267)153 x (66666)72 = unit's digit in (7)153 x (6)72 = unit's digit in 7 x 6 = 2 Hence, D (2) is the correct answer.
 Question 14
What is the total number of prime factors in the expression 1212 x 1616 x 1818 ?
 A 46 B 154 C 3456 D 2
Numbers
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Question 14 Explanation:
1212 = (2 x 2 x 3)12 = 224 x 312 1616 = (2 x 2 x 2 x 2)16 = 264 1818 = (2 x 3 x 3)18 = 218 x 336   Therefore, total number of prime factors = 24 + 12 + 64 + 18 + 36 = 154
 Question 15
What should be assigned to # so that 2582#724 is divisible by 11 ?
 A 4 B 5 C 6 D 7
Numbers    Number Divisibility
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Question 15 Explanation:
For a number to be divisible by 11, the difference of the sum of numbers at even and odd places should be either 0 or a multiple of 11. We assume the leftmost digit at position 1. So, Sum of numbers at even places = 5 + 2 + 7 + 4 = 18 Sum of numbers at odd places = 2 + 8 + # + 2 = 12 + #   Now, to make the number divisible by 11, we equate the sums obtained in the above step. 18 = 12 + # => # = 6 Therefore, C (6) is the correct choice.
 Question 16
What must be subtracted from the greatest five digit number so as to find the greatest five digit number divisible by 23 ?
 A 22 B 20 C 18 D 16
Numbers    Number Divisibility
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Question 16 Explanation:
99999 is the greatest five digit number. 23 x 4347 = 99981, which is the largest five digit number divisible by 23. => 99999 - 99981 = 18 Thus, C (18) is the correct choice.
 Question 17
On dividing 201098 by a certain number, the quotient is 67 and the remainder is 31. Find the divisor.
 A 3011 B 3001 C 3021 D 2991
Numbers
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Question 17 Explanation:
Dividend = (Divisor x Quotient) + Remainder => Divisor = (Dividend - Remainder) / Quotient => Divisor = (201098 - 31) / 67 = 3001 Thus, B (3001) is the correct choice.
 Question 18
A number leaves remainders 1, 4 and 7 on being divided successively by 3, 5 and 8 respectively. What is the product of the remainders when the order of divisors is reversed ?
 A 28 B 48 C 24 D 36
Numbers
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Question 18 Explanation:
8 x 1 + 7 = 15 5 x 15 + 4 = 79 3 x 79 + 1 = 238 Thus, 238 is the number.   Now, we divide 238 successively by 8, 5, 3 in that particular order. 238 / 8 => Q = 29, R = 6 29 / 5 => Q = 5, R = 4 5 / 3 => Q = 1, R = 2 Therefore, product of remainders = 6 x 4 x 2 = 48
 Question 19
What is the smallest number which when divided by 6, 9, 11, 16 and 22 leaves remainder 3 in each case ?
 A 3267 B 1584 C 1587 D 9504
Numbers
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Question 19 Explanation:
LCM (6,9,11,16,22) = 1584 The required number = 1584 + 3 = 1587
 Question 20
What is the largest number which divides 26, 47, 84 to leave remainders 2, 3, 0 respectively ?
 A 4 B 6 C 8 D 1
Numbers
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Question 20 Explanation:
26 - 2 = 24 47 - 3 = 44 84 - 0 = 84 The number divides 26, 47, 84 to leave remainders 2, 3, 0 respectively. => The number should divide 24, 44, 84 completely.   HCF (24, 44, 84) = 4 Hence, 4 is the required number.
 Question 21
Three friends started running together on a circular track at 8:00:00 am. Time taken by them to complete one round of the track is 15 min, 20 min, 30 min respectively. If they run continuously without any halts, then at what time will they meet again at the starting point for the fourth time ?
 A 8:30:00 am B 9:00:00 pm C 12:00:00 pm D 12:00:00 am
Numbers    LCM
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Question 21 Explanation:
LCM (15, 20, 30) = 60 => They meet at the starting point after every 60 min, i.e., after every 1 hour. Therefore, they will meet at the starting point for the fourth time after 4 hours, i.e., at 12:00:00 pm.
 Question 22
The LCM of two co-prime numbers is 117. What is the sum of squares of the numbers ?
 A 220 B 1530 C 250 D 22
Numbers    LCM
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Question 22 Explanation:
117 = 3 x 3 x 13 As the numbers are co-prime, HCF = 1. So, the numbers have to be 9 and 13. 92 = 81 132 = 169 Therefore, required answer = 250
 Question 23
HCF of two numbers is 11 and their LCM is 385. If the numbers do not differ by more than 50, what is the sum of the two numbers ?
 A 132 B 35 C 12 D 36
Numbers    LCM    HCF
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Question 23 Explanation:
Product of numbers = LCM x HCF = 11 x 385 = 4235 Let the numbers be of the form 11m and 11n, such that 'm' and 'n' are co-primes. => 11m x 11n = 4235 => m x n = 35 => (m,n) can be either of (1, 35), (35, 1), (5, 7), (7, 5). => The numbers can be (11, 385), (385, 11), (55, 77), (77, 55). But it is given that the numbers cannot differ by more than 50. Hence, the numbers are 55 and 77. Therefore, sum of the two numbers = 55 + 77 = 132
 Question 24
Two numbers are in the ratio of 5:7. If their LCM is 105, what is the difference between their squares ?
 A 216 B 210 C 72 D 840
Numbers    LCM    HCF
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Question 24 Explanation:
Let 'h' be the HCF of the two numbers. => The numbers are 5h and 7h. We know that Product of Numbers = LCM x HCF => 5h x 7h = 105 x h => h = 3 So, the numbers are 15 and 21. Therefore, difference of their squares = 212 - 152 = 441 - 225 = 216
 Question 25
Which is the largest number that divides 17, 23, 35, 59 to leave the same remainder in each case ?
 A 2 B 3 C 6 D 12
Numbers    HCF
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Question 25 Explanation:
Required Number = HCF (23-17, 35-23, 59-35, 59-17) Required Number = HCF (6, 12, 24, 42) = 6
 Question 26
Which of the following is the largest of all ? (i) 7/8 (ii) 15/16 (iii) 23/24 (iv) 31/32
 A (i) B (ii) C (iii) D (iv)
Numbers    LCM    Number Divisibility
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Question 26 Explanation:
LCM (8, 16, 24, 32) = 96 7/8 = 84/96 15/16 = 90/96 23/24 = 92/96 31/32 = 93/96 Hence, 31/32 is the largest of all.
 Question 27
A partial order P is defined on the set of natural numbers as follows. Here x/y denotes integer division.
i. (0, 0) ∊ P.
ii. (a, b) ∊ P if and only if a % 10 ≤ b % 10 and (a/10, b/10) ∊ P.
Consider the following ordered pairs:
i. (101, 22)
ii. (22, 101)
iii. (145, 265)
iv. (0, 153)
Which of these ordered pairs of natural numbers are contained in P?
 A (i) and (iii) B (ii) and (iv) C (i) and (iv) D (iii) and (iv)
Numbers    Gate IT 2007
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There are 27 questions to complete.

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