Least Common Multiples

In arithmetic and number theory, the least common multiple or lowest common multiple (LCM) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both of a and of b. Since it is a multiple, it can be divided by a and b without a remainder. A few word problems for least common multiples is given below.

(Source: Wikipedia)

Example of word problems for least common multiples:

Word problem 1:

Find the largest number of four digits which when divided by 5, 10, 15, it leaves a remainder 4 in each case.

Solution:

Step 1: Given numbers

5, 10, 15

Step 2: Find least common multiple of 5, 10, 15

Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40....

Multiples of 10 = 10, 20, 30, 40, 50, 60, 70, 80...

Multiples of 15 = 15, 30, 45, 60, 75, 90, 105....

From the list of multiples of 5, 10 and 15 the smallest common number in each list 30.

Therefore, the least common multiples of 5, 10 and 15 are 30.

Step 3: Find multiple of 30 which should be slightly less than five digit.

30 * 333 = 9990

So, 9990 is the largest number of four digit which is divisible by 5, 10, 15 and leaves a remainder 0.

Step 4: Find number which leaves a remainder 4.

To get remainder 4, we should add 4 to the obtained number.

Therefore, the required number is 9990 + 4 = 9994

Word problem 2:

Three children John, Nick and Shane run on a round  track. John takes 50 seconds, Nick takes 55 seconds and Shane takes 60 seconds to run a round. If all three of them start together at a point, when do they meet again?

Solution:

Step 1: Find least common multiple of 50, 55, 60

10 |       50       55       60

-----------------------------------------

5  |      5          55       6

-----------------------------------------

1           11       6

Least common multiple = 10 * 5 * 1 * 11 * 6 = 3300

Step 2: Solution

Therefore, they meet after 3300 seconds = 55 minutes.

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Homework of word problems for least common multiples:

1) Find the largest number of three digits which when divided by 7, 14, 28 it leaves a remainder 2 in each case.

2) Two children Joseph and Fleming run on a round track. Joseph takes 75 seconds and Fleming takes 80 seconds to run a round. If both of them start together at a point, when do they meet again?

Solutions:

1) 982

2) 1200 seconds

How to Find Number of Factors

In mathematics, factorization (also factorization in British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2) (x + 2). In all cases, a product of simpler objects is obtained.

How to find number of factor:example problems

Example 1:

Find all number of factors 50.
Solution:

50 = 1x50
= 2x25
= 5x10

So, the factors of 50 are 1, 2, 5, 10, 25 and 50.In the above example for how to find number of factors.

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Example 2:

Find all numbers of factors 80

Solution:

80 = 1x80
= 2x40
= 4x20
= 5x16
= 8x10

So, the factors of 80 are 1, 2,4,5,8,10,16,20,40 and 80.In the above example for how to find number of factors.

More explanation of how to find number of factors:-

Every number greater than 1 have atleast two factors: 1 and itself.

Example,

2 = 1 * 2

3 = 1 * 3

4 = 1 * 4

In two numbers are factors of another number are multiplied.

Note:- In 4 has some other factors besides 1 and 4:

4 = 1 * 4

4 = 2 * 2

A number 36 is a factors 4,

36 =  1 * 36

=  2 * 18

=  3 * 12

=  4 *  9

=  6 *  6

In those factors 1 and 36:

36 = 1 * 36

Divide by the next highest number after 1 and 2 is goes to 36. 18 times, 2 and 18 are a pair of factors:

36 =  1 * 36

=  2 * 18

=  3 * 12

=  4 *  9

Note:- 5 doesn't work, so leave that, and go on to 6:

36 =  1 * 36

=  2 * 18

=  3 * 12

=  4 *  9

=  6 *  6

Again, 7 doesn't work, and neither does 8. But 9 works:

36 =  1 * 36

=  2 * 18

=  3 * 12

=  4 *  9

=  6 *  6

=  9 *  4

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but it would be add on the small  table, because it says the same thing as an earlier entry:

36 = 1 * 36

= 2 * 18

= 3 * 12

= 4 *  9  <-- p="">
= 6 *  6     |  the same factors

= 9 *  4  <-- p="">
Any numbers larger than 6 remaining are smaller than 6. The factors of 36 are 1, 36, 2, 18, 3, 12, 4, 9, and 6.

Numerical Integrals

The process of finding an integral; it may be definite integral or an indefinite integral is referred as integration. Numerical integration is mainly used for finding the numerical value of a definite integral. Numerical integration is also used to finding the numerical solution of differential equations. Numerical integration is also referred to as numerical quadrature.

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Numerical integrals:- types of integrals

1. Indefinite integrations:

An indefinite integration is the family of functions that have a given function as a common derivative. The indefinite integral of f(x) is written ∫ f(x) dx.

2. Definite integrations:

If F(x) is the integral of function f(x) over the interval [a, b] ,i.e., ∫ f(x) dx = F(x) then the definite integral of function  f(x) over the interval [a, b] is denoted by int_a^bf(x)dx and is defined as int_a^bf(x)dx  = F(b) – F(a).

Where 'a' is called the lower limit and b is called the upper limit of integration and the interval [a, b] is called of integration.

Methods of integration:

It is not possible to integrate each integral with help of the following methods but a large number of varieties of the problems can be solved by these methods so, we have the following methods of integration:

1. Integration by substitution

2. Integration by parts.

3. Partial fractions

Example Problems on indefinite and definite integration:

Numerical Integrals Problem:

Example 1:

find ∫ 1 / sin2x cos2x dx.

Solution:

We have ∫ 1/ sin2x cos2x dx

= ∫sin2x+cos2x / sin2x+cos2x dx

=∫ (1/cos2x+1/ sin2x )dx

=∫sec2 x d x+∫cosec 2 x dx

=tan x -cot x+C.

Example 2:

∫ sin x sin (cos x) dx.

Solution:

Let cos x = t

dt = -sin x dx

Therefore we have

∫sin x sin (cos x) dx = - ∫sint dt = cost + C=cos (cos x) + C.

where

t=cos x

Example 3:  int_0^1dx/(1+x2)

Solution:

=[tan-1 x]10

=tan-1 1 - tan-1 0

= /4 -0 = / 4

π/2

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Example 4: Find the value int sin⁷xdx

-π/2

Solution:

Let f (x) = dx.

Then f(-x) = -sin7 x=-f (x)

so, f (x) is odd function.

π/2

π/2

Therefore, int f(x)dx=0 => int sin7x dx = 0.

-π/2                  -π/2

Laws of Exponents for Real Numbers

Let ‘a’ be a positive number. Exponential notation is ax , where x is an integer. When x is a rational number, say p/q with p, an integer and q, a positive integer, we define ax by ax  = root(q)(a) p

Example:

53/8 = root(3)(5)8

laws of exponents for real numbers: Exponents Notation

When x is an irrational number, ax can be defined to represent a real number.

For any a > 0, ax can be defined and that it represents an unique real number u and write u = ax.

The real number u is written in the exponential form or in the exponential notation.

Here the positive number 'a' is called the base and x, the index or the power or the exponent.

The laws of indices which we have stated for integer exponents can be obtained for all real exponents.


laws of Exponents for Real Numbers

We state them here and call them, the laws of exponents for real number:

root(x)(root(y)(a))  = root(xy)(a)


root(x)(a / b)  =  root(x)(a) / root(x)(b)


(root(x)(a) )x = a


axxx ay = ax+y


ax// ay = ax-y


(ax)y = axy


a-x  = 1/ ax


axxx bx = abx


a0 = 1  for a != 0


(a/b)x = ax/bx


a1/x =  root(x)(a)


(root(x)(a) )y/x = (root(x)(a) )


root(x)(a) root(x)(b) = root(x)(ab)



Example sums for Exponents for Real Numbers

Example 1:

Find the exponential form of (25)1/17

Solution of example 1:

Given = (25)1/17

Exponential form  =  root(17)(25)

Example 2:

Simplify 25225 / 53253

Solution of example 2:

Given 25225 / 53253

Now we use the exponents formula for real numbers

By using

(i)axxx ay = ax+y

(ii)axxx bx = abx

We get

= 253 / (5(25))3

= 253 / 1253

= 253 / (5)3(25)3

Cancel the same variables

Now, we get

= 1 / 53

= 1 / 125

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Example 3:

Solve ( root(3)(root(2)(64))92xx 62) / ((43)2+(5/25)2)

Solution of example 3:

Given = ( root(3)(root(2)(64))92xx 62) / ((43)2+(5/25)2)

Now we use the exponents formula for real numbers

By using below laws of exponents formula we can simplify the given data.

(i) root(x)(root(y)(a)) = root(xy)(a)

(ii)axxx bx = abx

(iii)(ax)y = axy

(iv)(a/b)x = ax/bx

= ( root(3)(root(2)(64))92xx 62) / ((43)2+(5/25)2)

Use the (i) formula

We get

= root(6)(49))92xx 62) / ((43)2+(5/25)2)

Use the (ii) formula

We get

= ((root(6)(64))542) / ((43)2+(5/25)2)

Use the (iii) formula

We get

= ((root(6)(64))542) / ((46)+(5/25)2)

Use the (iii) formula

We get

= ((root(6)(64))542) / ((46)+(52/252))

= (2 (542) / ((256) + (25 / 252))

Cancel the same elements

We get

= 2(542) / ((256) + (1 / 25))

= 2(542) / ((6400 + 1) / 25 )

= 2(2916) / (6401/25)

= 2(2916)(25) / (6401)

= 50(2916) / 6401

= 145800 / 6401

= 22.77

What are Perpendicular lines

Two intersecting lines will have four angles formed at the intersection points. If all the four angles are equal, then the two lines are said to be perpendicular to each other. We already know by linear postulate theorem that the two vertically opposite angles are equal. Hence if these two lines are perpendicular, then all four angles are 90 degrees.

Examples of perpendicular lines:

In the graph paper, The X-axis and Y-axis are perpendicular.
In an ellipse two axes, minor axis, and major axis are perpendicular.
For a line segment, any shortest line from a point outside the circle is perpendicular.
Tangent and normal to any curve are perpendicular lines.

Slopes of two perpendicular lines:  In coordinate Geometry, when two lines are perpendicular, the product of the slopes of the lines is -1.  This property has a lot of applications in finding the equation of perpendicular lines, length of perpendicular segment from a point to a given line, etc.

For any curve in a graph with equation y = f(x), the slope of the tangent is defined as the rate of change of y with respect to x at that point. The normal to this curve at this point is perpendicular to the tangent line.

Example:  In a circle, with centre at the origin and radius 3, the equation will be of the form (x)²+(y)² = 3². Take any point say (0,3). To find the tangent, we have to find dy/dx.

Differentiating, 2x+2y  =0

Hence, the slope of the normal is perpendicular to x axis or parallel to y axis.

Example for Perpendicular lines from a point to a line

Let AB be a line with coordinates (1,2) and (3,4).  Measure the length of perpendicular line from (-1,1) to this line segment.

We know that the perpendicular line from (-1,1) has a slope  of -1/slope of AB.

Equation of AB is (x-1)/(3-1) = (y-2)/(4-2)  Or x-1 = y-2 Or y = x+1

Slope of AB passing through (1,2) and (3,4) is 4-2/3-1 =1.

Slope of perpendicular line to AB is -1.

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Since the perpendicular line passes through (-1,1) equation of the perpendicular is y-1 = -1(x+1)  or y =-x -1 +1 or y = -x.

To get the foot of the perpendicular line on AB, we solve the two equations by substitution method.

y = x+1 = -x   This on simplification gives 2x=-1 or x=-1/2.

Since y = -x , we have y = +1/2,

So, foot of the altitude from the point (-1,1) is (-1/2,1/2).

The length of the perpendicular segment is between (-1,1) and (-1/2,1/2) is

√[ (-1/2+1)²+(1/2-1)²] = √(1/4+1/4)   =  √(1/2) = 1/1.414 =0.707 approximately.

Examples of Concave Polygon

The concave polygon has single interior angle which is more than 180 degree. The concave method is drawing few of the straight line only. The concave polygon is used in many interior angles. The three sided polygon denoted the triangle, triangle cannot be a concave polygon. In this article we shall discuss the examples and properties of concave polygon.

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Properties of concave polygons:

The concave polygons do not calculate the exterior angle. The opposite meaning of the concave is called the convex. The polygon is normally declared as concave polygon, but not the convex polygon. The interior angle must be reflex angle of the concave polygon.

The diagonal of the concave polygon is the line through the outside of the polygon. The star polygon is the example of non simple polygon. It must  contain the concave polygon with minimum of four sides. The area of the concave polygon and irregular polygons are same.

Example of the concave polygon

The first example of the concave polygon is declaring the following figure. The concave polygon connected the line of the interior angle side. The important part is declaring interior. There are only  two points selected inside the diagram. Do not select the one point which is inside of the angle and another point is outside of the diagram.

diagram repsent the example of the concave polygon

The inside of the polygon does not enter the line P and Q. The line A and B is entering the inside of the polygon. So that it is said to be concave polygon. The minimum one line segment will not present inside the diagram, this part of the diagram is called the concave.


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The above example is commonly called the concave polygon. The next example of the concave polygon is declaring the following diagram. In figure the blue line segment is declared as two points enter into the inside of the diagram. That line through the outside of the diagram.

Learning Disjunction

Definition:

Logical disjunction is an operation on the two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are false. More generally a disjunction was a logical formula that can have one or more literals separated only by ORs. A single literal is frequently considered to be a degenerate disjunction.

Properties:

Associativity :

aV (bVc) = (aVb) V c

In mathematics, associativity was a property of some binary operations. It means that, within the expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. Consider for instance the equation

(5+2) + 1 = 5 + (2+1) = 8

Commutativity:

In mathematics, commutativity is the assets that changing the order of something does not change the end result. It is a primary property of many binary operations, and many mathematical proofs depend on it. The commutativity of easy operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the property was not named until the 19th century when mathematics started to become formalized.

Distributivity:

In mathematics, and in particular to abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra. For example:

2 × (1 + 3) = (2 × 1) + (2 × 3).

Idempotence:

Idempotence is a property of certain operations in mathematics and computer science. Idempotent operations are operations that can apply multiple times without changing the result. The conception of idempotence arises in a number of places in abstract algebra and functional programming.

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Monotonic function:

In mathematics, a monotonic function in which conserve the given order. This concept first arose in calculus, and was shortly generalized to the more abstract setting of order theory.

Symbol:

The mathematical symbol for logical disjunction varies in the text. In addition to the word "or", the symbol "V", deriving from the Latin word vel for "or", is commonly used for disjunction. For example: "A V B " is read as "A or B ". Such a disjunction is false if both A and B are false. In all other cases it is true.

All of the following are disjunctions:

A V B

¬A V B

A V ¬B V ¬C V D V ¬E