LCM (Least Common Factor)

LCM means Least Common Factor

Find the LCM of 12, 18 and 24.

Method 1:

LCM = 2³ × 3² = 72

Method 2:

12 = 2² × 3

18 = 2 × 3²

24 = 2³ × 3

LCM = 2³ 2³ × 3² = 72

LCM is the product of highest power of each distinct factor.

Exercise:

1. Find the least number exactly divisible by 10, 12, 15, and 24.
2. Find the largest 3-digit number exactly divisible by 10, 12, 15, and 24.
3. Find the least 4-digit number exactly divisible by 10, 12, 15, and 24.
4. Find the least number which when divided by 12, 15, 20 and 27 leaves remainder 2 in each case.
5. Find the least number which when divided by 52 leaves remainder 35, when divided by 78 leaves remainder 61 and when divided by 117 leaves remainder 100.
6. What smallest number can be subtracted from 1000, so that remainder may be divisible by 24, 36 and 48?
7. Find the least square number which is divisible by 3, 4, 5, 6 and 8.
8. There are 5 bells which starts ringing simultaneously together at interval of 3, 6, 9, 12 and 15 seconds. In 36 minutes, how many times the bells will ring simultaneously?
9. Two men started moving on circular path from same place at same time in same direction. If they complete one rotation in 10 and 15 minutes respectively. After how long they will meet again?

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Simplification

Rule: The simplification is done in following order: Bracket, off, division, multiplication, addition and at last subtraction. It is famously called as BODMAS.

BODMAS

B -> bracket. It can be [], {}, ()

O -> off

D -> division

M -> multiplication

A -> addition

S -> subtraction

The simplification is done in order of BODMAS.

Exercise

Find the value of 1 ÷ [ 1 + ÷ { 1 + 1 ÷ ( 1 + 1 ÷ 2) } ].

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Concept of fraction

Fraction is a number in the form of a/b.
Here ‘a’ is called numerator, ‘b’ is called denominator.

For example: ½, ¾. 

Types of fraction:

1. Proper fraction: In proper fraction numerator is less than denominator i.e.
   numerator < denominator
   For Example: 2/5, 5/8…
2. Improper fraction:In improper fraction numerator is greater than denominator i.e.
   For Example: 5/2, 8/5…
3. Mixed fraction:It is a combination of whole number and proper fraction.
   For example: 2½, 3½….

Comparison of fractions

1. Find the largest from the following:11/13, 9/11
   Trick: do cross multiplication11×11 > 13×9
   Clearly 11/13 is greater than 9/11.
2. When difference between numerator and denominator is same
1. If fraction is proper:In this case, the fraction having bigger numerator is greater.
   For example: 15/18, 7/10, 11/14, 8/11
   15/18 > 11/14 > 8/11 > 7/10
2. If fraction is improper:In this case, the fraction having smaller numerator is smaller.
   For example: 11/8, 8/5, 31/28, 17/14
   8/5 > 11/8 > 17/14 > 31/28

Addition of mixed fraction

Example: 6 ²/₃ + 3 ¹/₃ + 5 ¹/₆
                = (6 + 3 +5) + (2/3 + 1/3 + 1/6) = 15 ¹/₆

Exercise:

1. Find the value of 6 ²/₃ - 5 ¹/₆ - 4 ²/₃
2. Find the value of 999 ¹/₇ + 999 ²/₇ + 999 ³/₇ + 999 ⁴/₇ + 999 ⁵/₇ + 999 ⁶/₇.

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Surds

If ‘a’ be real number and ‘n’ be positive integer such that a^1/n is an irrational then a^1/n is called surd of order ‘n’.

Rules:

1. ⁿ√a = a^1/n
2. ⁿ√ab = ⁿ√a × ⁿ√b
3. ⁿ√a/b = ⁿ√a / ⁿ√b
4. (ⁿ√a)ⁿ = a
5. ^m√ⁿ√a = ^mn√a
6. (ⁿ√a)^m = ⁿ√a^m

Exercise

1. Find the value of (36)^1/6.
2. Find the value of √50 + √98.
3. Find the value of 2³√40 - 4³√320 + 3³√625 - 3³√5
4. Find the value of ³√2 × ⁶√3 × √5.
5. Find the smallest among the following:
   ³√3, ³√2, ²√2, ³√4
6. Find the largest among the following:
   ³√4, ⁴√6, ⁶√15, ¹²√245

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Indices

Rule:

1. a^m × aⁿ = a^m+n
2. a^m / aⁿ = a^m-n
3. (a^m)ⁿ = a^mn
4. (ab)^m = a^mb^m
5. (a/b)^m = a^m/b^m
6. a⁰ = 1
7. if a^m = aⁿ, then m = n
8. a^m = k, then a = k^1/m

Exercise

1. Find the value of 256^0.16 × 256^0.9?
2. Find the value of (0.01024)⁵?
3. Find the value of (x^a/x^b)^1/ab × (x^b/x^c)^1/bc × (x^c/x^a)^1/ca ?
4. If 3^x+8 = 27^2x+1, then find the value of x.
5. If x = 10²⁴, y = 10⁷², x^z = y² then find the value of z.
6. If a^x = b^y = c^z and b² = ac, then find the value of y.

Some tips in number system

1. The sum of five consecutive whole numbers is always divisible by 5.
2. The product of three consecutive natural numbers is always divisible by 6.
3. The product of three consecutive natural numbers, the first of which is even number is always divisible by 24.
4. For any natural number ‘n’, n³ – n is divisible by 6.
5. Any number written in the form of 10ⁿ – 1, is divisible by 3 or 9, where ‘n’ is natural number.
6. The square odd number when divided by 8 leaves remainder 1.
7. For any prime number P > 3, P² – 1 is divisible by 24.
8. aⁿ + bⁿ is always divisible by a + b, when ‘n’ is odd.
9. aⁿ + bⁿ is never divisible by a – b.
10. aⁿ – bⁿ is exactly divisible by a + b, when ‘n’ is even.
11. aⁿ – bⁿ is exactly divisible by a – b, when ‘n’ is even or odd.
12. The sum of two digit number and number formed by interchanging its digits is always divisible by 11.
13. The difference of two digit number and number formed by interchanging its digits is always divisible by 9

Find number of zeroes at the end of product

Rule:

Convert the number in the form of 2^m × 5ⁿ

1. Number of zeroes = m, if m < n
2. Number of zeroes = n, if n < m

Problems:

1. Find the number of zeroes at the end of 24 × 32 × 17 × 13
   Solution: 24 × 32 × 17 × 13
   = 2³ × 3 × 2⁵ × 17 × 13
   As there is no 5 in it, so there will be no 0 in this product.
2. Find the number of zeroes at the end of 8 × 15 × 25 × 22 × 13 × 19
   Solution: 8 × 15 × 25 × 22 × 13 × 19
   = 2³ × 3 × 5 × 5² × 2 × 11 × 13 × 19
   = 2⁴ × 5³ × 3  × 11 × 13 × 19
   as there are power of 5 is lesser i.e. 3 so no of zeroes will be 3.

Concept of Unit Digit

To find the unit digit in (xyz)ⁿ

Here xyz is some number, n is the Index and z is the base unit.

We need to find the unit digit of (xyz)ⁿ

Rule:
Divide the index ‘n’ by 4, the remainder (R) can be 0, 1, 2 or 3.

1. If R = 0, then
• If base unit is odd except 5, then the unit digit will be 1.
• If base unit is even, then the unit digit will be 6.
2. If R = 1, 2 or 3, then
   unit digit =  unit digit of (base unit)^R

Note: if base unit is 0, 1, 5 or 6, the unit digit will be same as base unit.

Problems:

1. Find the unit digit of (1237)¹³²132/4 leaves remainder 0.
   As base unit is odd, the unit digit of (1237)¹³² is 1.
2. Find the unit digit of (1238)¹³²132/4 leaves remainder 0.
   As base unit is even, the unit digit of (1238)¹³² is 6.

Exercise:

1. Find the unit digit of 1234¹³² × 5247²⁵⁶ × 1353¹³³?
2. Find the unit digit of 1234¹³² + 5247²⁵⁶ + 1353¹³³?

Concept of Remainder

Hello guys! Today we will discuss about concept of remainder and other methods to find remainder of a number.

Firstly tell me what will be the remainder when number 22 is divided by 5. Yes, as you guess it will be 2 as depicted below:

Now what will be the remainder when -22 is divided by 5? Many of you might guess it will be -2, but you are wrong. So let’s see how:

It is 3.

Note: Remainder is always non-negative. It cannot be negative.

Rule: Dividend = Divisor × Quotient + Remainder

Remainder Theorem

It states that remainder  of product of number is such that:

Where a_r = remainder when a is divided by n,
             b_r = remainder when b is divided by n,
             c_r = remainder when c is divided by n.

Problems:

1.       Find remainder of (32×28×21)/6 

       (32×28×21)/6 à (2×4×3)/6 à24/6 à 0

       Remainder is 0.

2.       Find remainder of (51×61×71)/4 

       (51×61×71)/4 à (3×1×3)/4 à9/4 à 1 

       Remainder is 1.

Similarly for addition, the method for calculating the remainder is same as given below:

Problems:

1. Find remainder of (32+28+21)/6
   (32+28+21)/6 à (2+4+3)/6 à9/6 à 3
   Remainder is 3.
2. Find remainder of (51+61+71)/4
   (51+61+71)/4 à (3+1+3)/4 à7/4 à 3
   Remainder is 3.

Important Rules to find the remainder

Rule 1:  For ‘a’ and ‘n’ be any position integer

1. Remainder of aⁿ/(a+1) is ‘a‘ i.e.
   aⁿ/(a+1) à a, if ‘n’ is odd.
2. Remainder of aⁿ/(a+1) is ‘1‘ i.e.
   aⁿ/(a+1) à 1, if ‘n’ is even.

Problems:

1. Remainder of 2¹⁰⁰/3 = 1.
2. Remainder of 3⁹⁹/4 = 3.
3. Remainder of 65²⁰⁵/66 = 65.
4. Find remainder of (65²⁰⁶+1)/66
   (65²⁰⁶+1)/66 à (1+1)/66 à2

Rule 2: For ‘a’ and ‘n’ be any position integer

1. Remainder of (ax+b)ⁿ/a is remainder of bⁿ/a
   (ax+b)ⁿ/a àremainder of bⁿ/a
2. Remainder of (ax+1)ⁿ/a is 1
   (ax+1)ⁿ/a àremainder of bⁿ/a

Problems:

1. Find the remainder of 51²⁰³/7
   Solution: 51²⁰³/7 = (7×7+2)²⁰³/7 à 2²⁰³/7 = ((2³)⁶⁷×2²)/7 à (8⁶⁷×4)/7 à ((7×1+1)⁶⁷×4)/7 à (1×4)/7 à 4

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Different type of numbers

·         Natural Number (N): it is the set of counting numbers i.e. {1, 2, 3…..}

·         Whole Number (W): It is the set of counting numbers including zero in the Natural numbers.
i.e. {0, 1, 2, 3….}

·         Integer (Z): It is the set of natural numbers with zero and negative natural numbers. i.e. {….-3, -2, -1, 0, 1, 2, 3…}

·         Rational Number (Q): The numbers which can be represented in the form p/q where p, q are integers and q ≠ 0
Ex: 1/2, 3/4, 1/2…

·         Irrational Numbers: The numbers which are not rational. These numbers cannot be written as simple fraction.
Ex: π, √2 ,√3 …

·         Real Number (R): The set of numbers which includes both rational and irrational numbers.
Real numbers = Rational number + Irrational numbers + Whole numbers.
It can be positive, negative or zero.

·         Prime Number: It is number which have no factor other than itself and unity.
Ex: 2, 3, 5, 7, 11, 13….
Note: 2 is the only even number which is prime.

·         Composite Number: The numbers which are not prime are called composite numbers.
Ex: 4, 6, 8, 9, 10…

·         Perfect Number: It is a number which is equal to the sum of all of its divisors excluding itself.
Ex: 6 = 1 + 2 + 3
    28 = 1 + 2 + 4 + 7 + 14

·         Absolute value of number: It is denoted by |x|
|x| = x if x is positive
       = -x if x is negative
Note: |x| is always positive.
Ex: |-6| = 6, |20| = 20 etc.

Number System

Questions from number system are predominant in SSC. So let’s get started with it. 

We are going to learn following topics in Number system.

1. Different type of numbers
2. Concept of Remainder
3. Concept of Unit digit
4. Find number of zeroes at the end of product
5. Some tips in number system
6. Indices
7. Surds
8. Concept of fraction
9. Simplification
10. LCM