Numbers Aptitude basics, practice questions, answers and explanations Prepare for companies tests and interviews

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 = ap . bq . cr ….. 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 = (ap+1 – 1 / a – 1) . (bq+1 – 1 / b – 1) . (cr+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 2n-1, where ‘n’ is the number of different prime factors of the given number N.