Numbers
Numbers
Important Formulas
Some Basic Formulae:
 (a + b)(a – b) = (a^{2} – b^{2})
 (a + b)^{2} = (a^{2} + b^{2} + 2ab)
 (a – b)^{2} = (a^{2} + b^{2} – 2ab)
 (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2(ab + bc + ca)
 (a^{3} + b^{3}) = (a + b)(a^{2} – ab + b^{2})
 (a^{3} – b^{3}) = (a – b)(a^{2} + ab + b^{2})
 (a^{3} + b^{3} + c^{3} – 3abc) = (a + b + c)(a^{2} + b^{2} + c^{2} – ab – bc – ac)
 When a + b + c = 0, then a^{3} + b^{3} + c^{3} = 3abc.
1
Which one of the following is not a prime number?  

View Answer
Answer: Option D
Explanation:
91 is divisible by 7. So, it is not a prime number.
2.
(112 x 5^{4}) = ?  

View Answer
Answer: Option B
Explanation:
(112 x 5^{4}) = 112 x  10  4  =  112 x 10^{4}  =  1120000  = 70000  
2  2^{4}  16 
3.
View Answer
Answer: Option D
Explanation:
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.
4.
What least number must be added to 1056, so that the sum is completely divisible by 23 ?  

View Answer
Answer: Option A
Explanation:
23) 1056 (45
92

136
115

21

Required number = (23  21)
= 2.
5.
1397 x 1397 = ?  

View Answer
Answer: Option A
Explanation:
1397 x 1397  = (1397)^{2} 
= (1400 – 3)^{2}  
= (1400)^{2} + (3)^{2} – (2 x 1400 x 3)  
= 1960000 + 9 – 8400  
= 1960009 – 8400  
= 1951609. 
6.
How many of the following numbers are divisible by 132 ? 264, 396, 462, 792, 968, 2178, 5184, 6336 


View Answer
Answer: Option A
Explanation:
132 = 4 x 3 x 11
So, if the number divisible by all the three number 4, 3 and 11, then the number is divisible by 132 also.
264 11,3,4 (/)
396 11,3,4 (/)
462 11,3 (X)
792 11,3,4 (/)
968 11,4 (X)
2178 11,3 (X)
5184 3,4 (X)
6336 11,3,4 (/)
Therefore the following numbers are divisible by 132 : 264, 396, 792 and 6336.
Required number of number = 4.
7.
The largest 4 digit number exactly divisible by 88 is:  

View Answer
Answer: Option A
Explanation:
Largest 4digit number = 9999
88) 9999 (113
88

119
88

319
264

55

Required number = (9999  55)
= 9944.
8.
What is the unit digit in {(6374)^{1793} x (625)^{317} x (341^{491})}?  

View Answer
Answer: Option A
Explanation:
Unit digit in (6374)^{1793} = Unit digit in (4)^{1793}
= Unit digit in [(4^{2})^{896} x 4]
= Unit digit in (6 x 4) = 4
Unit digit in (625)^{317} = Unit digit in (5)^{317} = 5
Unit digit in (341)^{491} = Unit digit in (1)^{491} = 1
Required digit = Unit digit in (4 x 5 x 1) = 0.
9.
The sum of first five prime numbers is:  

View Answer
Answer: Option D
Explanation:
Required sum = (2 + 3 + 5 + 7 + 11) = 28.
Note: 1 is not a prime number.
Definition: A prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself.
10.
The difference of two numbers is 1365. On dividing the larger number by the smaller, we get 6 as quotient and the 15 as remainder. What is the smaller number ?  

View Answer
Answer: Option B
Explanation:
Let the smaller number be x. Then larger number = (x + 1365).
x + 1365 = 6x + 15
5x = 1350
x = 270
Smaller number = 270.
11
If the number 517*324 is completely divisible by 3, then the smallest whole number in the place of * will be:  

View Answer
Answer: Option C
Explanation:
Sum of digits = (5 + 1 + 7 + x + 3 + 2 + 4) = (22 + x), which must be divisible by 3.
x = 2.
12.
The smallest 3 digit prime number is:  

View Answer
Answer: Option A
Explanation:
The smallest 3digit number is 100, which is divisible by 2.
100 is not a prime number.
101 < 11 and 101 is not divisible by any of the prime numbers 2, 3, 5, 7, 11.
101 is a prime number.
Hence 101 is the smallest 3digit prime number.
13.
Which one of the following numbers is exactly divisible by 11?  

View Answer
Answer: Option D
Explanation:
(4 + 5 + 2) – (1 + 6 + 3) = 1, not divisible by 11.
(2 + 6 + 4) – (4 + 5 + 2) = 1, not divisible by 11.
(4 + 6 + 1) – (2 + 5 + 3) = 1, not divisible by 11.
(4 + 6 + 1) – (2 + 5 + 4) = 0, So, 415624 is divisible by 11.
14.
(?) – 19657 – 33994 = 9999  

View Answer
Answer: Option A
Explanation:
19657 Let x  53651 = 9999
33994 Then, x = 9999 + 53651 = 63650

53651

15.
The sum of first 45 natural numbers is:  

View Answer
Answer: Option A
Explanation:
Let S_{n} =(1 + 2 + 3 + … + 45). This is an A.P. in which a =1, d =1, n = 45.
S_{n} =  n  [2a + (n – 1)d]  =  45  x [2 x 1 + (45 – 1) x 1]  =  45  x 46  = (45 x 23)  
2  2  2 
= 45 x (20 + 3)
= 45 x 20 + 45 x 3
= 900 + 135
= 1035.
Shorcut Method:
S_{n} =  n(n + 1)  =  45(45 + 1)  = 1035. 
2  2 