Numbers Aptitude basics, practice questions, answers and explanations 
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Divisibility

1. A number is divisible by 2 if it is an even number.

2. A number is divisible by 3 if the sum of the digits is divisible by 3.

3. A number is divisible by 4 if the number formed by the last two digits is divisible by 4.

4. A number is divisible by 5 if the units digit is either 5 or 0.

5. A number is divisible by 6 if the number is divisible by both 2 and 3.

6. A number is divisible by 8 if the number formed by the last three digits is divisible by 8.

7. A number is divisible by 9 if the sum of the digits is divisible by 9.

8. A number is divisible by 10 if the units digit is 0.

9. A number is divisible by 11 if the difference of the sum of its digits at odd

places and the sum of its digits at even places, is divisible by 11.


Important formulas

i. ( a + b )( a - b ) = ( a ² - b² )

ii. ( a + b ) ² = ( a² + b² + 2 ab )

iii. ( a - b )² = ( a² + b² - 2 ab )

iv. ( a + b + c ) ² = a²+ b² + c² + 2 ( ab + bc + ca )

v. ( a³ + b³ ) = ( a + b )( a² - ab + b² )

vi. ( a³ - b³ ) = ( a - b )( a² + ab + b²)

vii. Sum of natural numbers from 1 to n
      
                   
viii. Sum of squares of first n natural numbers is =

                    
ix. Sum of cubes of first n natural numbers is A

                                  
x. HCF= (HCF of the numerators)/(LCM of the denominators)

xi. LCM= (LCM of the numerators)/HCF of the denominators

xii. Product of two numbers = Product of their H.C.F. and L.C.M

 

Note: When a number N is raised to any integral power m, the digit in the unit’s

place of the resulting value can be determined without actually evaluating the

power. The digits when raised to powers will give values in which the digits in

the unit’s place follow a cylindrical pattern. Following is the pattern to calculate

the digit in the unit’s place of any derived power.

                                      

HCF models:-

If N is a composite number such that N = a^p . b^q . c^r ….. where a, b, c are prime factors of N and p,q,r ….. are positive integers, then


(a) The number of factors of N is given by the expression (p + 1) (q + 1) (r + 1)…


(b) It can be expressed as the product of two factors in 1/2 {(p + 1) (q + 1) (r + 1)…..}ways


(c) If N is a perfect square, it can be expressed


    (i)as a product of two DIFFERENT factors in 1/2 {(p + 1) (q + 1) (r + 1)…..   -1}ways


    (ii)as a product of two factors in 1/2 {(p + 1) (q + 1) (r + 1) ….+1}ways


(d) Sum of all factors of N = (a^p+1 – 1 / a – 1) . (b^q+1 – 1 / b – 1) . (c^r+1 – 1 / c – 1)…..


(e) The number of co-primes of N (< N),Ø(N) = N(1 – 1/a) (1 – 1/b) (1 – 1/c) ….


(f) Sum of the numbers in (e) = N/2 . Ø(N)


(g) It can be expressed as a product of two factors in 2^n-1, where ‘n’ is the number of different prime factors of the given number N.



Exercise Questions

1. 117 * 117 + 83 * 83 = ?


a) 20698

b) 20578

c) 21698

d) 21268


2. (1/4)3 + (3/4)3 + 3(1/4)(3/4)(1/4 + 3/4) =?


a) 1/64

b)27/64

c) 49/64

d)1


3. The difference of two numbers is 1365. On dividing the larger number by the smaller, we get 6 as quotient and 15 as reminder. What is the smaller number ?


a) 240

b) 270

c) 295

d) 360


4. The 7th digit of (202)³ is

a) 2
b) 4
c) 8
d) 6


5. H.C.F. of two numbers is 16. Which one of the following can never be their L.C.M

a) 32

b) 80

c) 64 

d) 60


6. What is the remainder when 9^1+ 9^2 + 9^3 + .... + 9^8 is divided by 6?

a) 3
b) 2
c) 0
d)5


7. The sum of the first 100 natural numbers is divisible by

a) 2, 4 and 8

b) 2 and 4

c)2 only

d)none of these


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

a) 2

b)3

c)6

d)4


9. 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?

a) 144 
b) 168
c)192 
d)none of these


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

a) 99840
b) 99900 
c)99960
d) 99990


Answer Key:

1.B; 2.D; 3.B; 4.C; 5.D; 6.C; 7.C; 8.B; 9.C; 10.C