Quantitative Aptitude - SPLessons

Chapter 1

Numbers Problems

Numbers Problems

Number System is all about finding the face value & place value of a digit, basic rules and different types of numbers, rules for divisibility, factors and multiples.


Types of numbers: There are various types of numbers. They are natural numbers, whole numbers, integers, even numbers, odd numbers, prime numbers, composite numbers, etc.

  1. Natural numbers are defined as the numbers that occur commonly in nature. A natural number is a whole, non negative number and the set of natural numbers is denoted by letter ‘N’.
    Set of natural numbers is N = {1,2,3…..}
  2. Whole numbers are defined as all the numbers without fractions and no decimals.
    Set of whole numbers is (W)={0,1,2,3…….}
    Note: Every natural number is a whole number except zero is a whole number which is not a natural number.
  3. Integers are defined as all the numbers i.e. zero, positive and negative numbers.
    Set of integers = {……-3,-2,-1,0,1,2,3,…….}
  4. Even numbers are defined as the numbers divisible by 2. i.e When an even number is divided by 2, the remainder is 0.
    0,2,4,6,….. are even numbers.
  5. Odd numbers are defined as the numbers which are not divisible by 2. i.e When an even number is divided by 2, the remainder is 1.
    1,3,5,7,…… are odd numbers.
  6. Prime numbers are those numbers which have exactly two factors namely itself and 1.
    Example: 2,3,5,7,11,13,17….etc.
  7. Composite numbers are defined as the numbers which are not prime.
    Example: 4,6,8,12,15,….. etc.

Rules for divisibility: 

  1. Divisibility by 2: A number is divisible by 2 if the last digit in a given numerical value is 0,2,4,6,8.
    Example: 526492 is divisible by 2.    (as last digit is 2)
  2. Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
    Example: 562185 is divisible by 3.    (as sum of digits =27 is divisible)
  3. Divisibility by 4: If the number formed by last two digits is divisible by 4 then the given numerical value is also divisible by 4.
    Example: 1459872 is divisible by 4.  (as last two digits 72 is divisible by 4)
  4. Divisibility by 5: If the units digit is 0 or 5, then the given number is divisible by 5.
    Example: 256480 is divisible by 5.   (as units digit is 0)
  5. Divisibility by 8: If the number formed by last three digits is divisible by 8 then the given numerical value is also divisible by 8.
    Example: 45566568 divisible by 8.    (as last three digits is divisible by 8)
  6. Divisibility by 9: If the sum of its digits is divisible by 9 then the given numerical value is also divisible by 9.
    Example: 21421899 divisible by 9.   (as sum of its digits = 36 is divisible by 9)
  7. Divisibility by 10: If the units digit is 0, then the given number is divisible by 10.
    Example: 65712590 is divisible by 10.    (as units digit is 0)
  8. Divisibility by 11: If the difference between sum of its digits at odd places and sum of its digits in even places is either zero or a number divisible by 11.
    Example: 25784 is divisible by 11.     (as (2+ 7 + 4) – (5+8) = 13 – 13 =0)

  • (Divisor × Quotient) + Remainder = Dividend
  • \((a+b)^2 = a^2 + b^2 +2ab\)
  • \((a-b)^2 = a^2 + b^2 -2ab\)
  • \(a^2 – b^2 = (a+b)(a-b)\)
  • \((1+2+3+….+n) = \frac{1}{2n(n+1)}\)
  • \(1^2+2^2+3^2+….+n^2 = \frac{1}{6n(n+1)(2n+1)}\)
  • \(1^3+2^3+3^2+….+n^3 = \frac{1}{4n^2(n+1)^2}\)
  • For Arithmetic Progression
    (i) nth term = \(a + (n – 1)d\)
    (ii) sum of n terms = \(\frac{n}{2(2a+(n-1)d)}\)
    (iii) sum of n terms = \(\frac{n}{2(a+l)}\), where l is the last term
  • For Geometric Progression
    (i) nth term = \(ar^{(n-1)}\)
    (ii) sum of n terms = \(\frac{a(1-r^n)}{(1-r)}\) when r<1 ; \(\frac{a(r^n-1)}{(r-1)}\) when r>1

Model 1: Find the unknown number 7429-?=4358-1587
Solution:
Take the unknown value as \(x\)
by substituting \(x\),
7429 – \(x\) = 4358 – 1587
7429 – \(x\) = 2771
\(x\) = 7429 – 2771
\(x\) = 4658
Therefore, the unknown value \(x\) = 4658

Model 2: Face value of 7 and Place value of 9 from the given digit 38745962 is?
Solution:

As, Face value is the value of digit itself and should relate place with ‘place value’
Given digit is 38745962
Therefore, face value of 7 is 7
(100 is multiplied since 9 is in hundred’s place)
Place value of 9 is (9 × 100)=900

Model 3: On dividing 132 by a certain number, 12 as a quotient and 0 as a remainder is obtained. Find the divisor?
Solution:

Given that Dividend = 132, Quotient = 12, Remainder = 0
[Divisor × Quotient] + Remainder=Dividend
by substituting the given values,
[Divisor × 12] + 0 = 132
Divisor = \(\frac{132}{12}\)
∴ Divisor = 11

Model 4: What will be the units digit in the product (234 × 256 × 457 × 952)?
Solution: 

Given product is (234 × 256 × 457 × 952)
Here units digits are 4,6,7,2
Therefore, product of Units digit in the given product = 4 × 6 × 7 × 2 = 336.

Model 5: Test which of the following is not a prime number a)19 b)11 c) 16 d) 13?
Solution:

As Prime number is the number which has exactly two factors namely itself
Factors of 19 = 1, 19
Factors of 11 = 1, 11
Factors of 16 = 1, 2, 4, 16
Factors of 13 = 1, 13
Therefore, 16 is not a prime as it is divisible by more than twice numbers.

Model 6: Does the number 23679715 divisible by 11?
Solution:

Given digit is 23679715
(Sum of digits at odd places)-(Sum of digits at even places)
Digits at odd places = 5, 7, 7, 3
Digits at even places = 1, 9, 6, 2
⇒ (5 + 7 + 7 + 3) – (1 + 9 + 6 + 2)
22 – 18 = 4
as 4 is not divisible by 11, 23679715 also not divisible by 11