Least Common Multiple of two Numbers

There are various mathematical concepts and terms that are to be learnt. The term LCM stands for least common multiple. This is an operation and can be carried out on two integers. The answer we obtain is the least common multiple. This denotes the least number that is fully divisible by both the integers.

Now how to find LCM is the question. It is a rather an easy task and can be performed if one is familiar with the basic arithmetic operations of mathematics and number theory. Division is the operation that is most frequently used. So, one must be confident in the division of two numbers.  So, to understand how to find the least common multiple one must be thorough be with the basic concepts in mathematics otherwise it would be really tough to proceed ahead.

So, what is LCM and how can be it found out. The numbers are selected first, for which it has to be found out. Then they are divided by a common number which should be able to divide all the numbers chosen completely. The number which is not divisible by the common number is written as it is for the next step to be performed. In the next again the numbers are divided by a number which is able to divide all the chosen numbers and this continues till ‘1’ is obtained at the end of all the steps of division. The numbers chosen are divided several times till ‘1’ is obtained in the end. Then the product of all the numbers which were used for division is obtained and this gives the least common multiple of the chosen numbers in the beginning.

So, what is the LCM of ‘3’ and ‘2’ is a question that must be answered and can explain the concept better. First a number which divides both ‘2’ and ‘3’ have to be found out. Since there is no common number that divides both ‘2’ and ‘3’ their product is found out. On multiplying ‘2’ and ‘3’, ‘6’ is obtained. So, ‘6’ is the least common multiple of ‘2’ and ‘3’. If two numbers are not divisible by a common number in the beginning itself, then their product itself is the least common multiple of the two. This is apt in the case of two prime numbers. In the example both ‘2’ and ‘3’ are prime numbers.

Multiplying Even and Odd Numbers

Even and odd numbers are two of the most important topics in basic arithmetic. All real numbers can be classified into even numbers and odd numbers. Even number is a number that can be evenly divided into two parts or can be exactly divided by 2. For example: I have to shed almost 12 kg post pregnancy weight . Here, 12 is an even number. On the other hand, odd number is a number that cannot be evenly divided into two parts or cannot be exactly divided by 2. For example: I have already shed 7 kg post pregnancy weight . Here, 7 is an odd number. There are some trend that can be seen while multiplying even and odd numbers. Let’s have a look at the multiplication rules for even and odd numbers in this post.

Multiplying Even Numbers
When two even numbers are multiplied, the result is always an even number. The formula is Even * Even = Even. For example: I need to list 6 different hobbies for kids for 2 magazines. 6 hobbies for kids * 2 magazines = 12. The result is an even number.

When an even number is multiplied by an odd number, the result is always even. The formula is Even * Odd = Even. For example: I have to survey minimum of 10 baby care center across 5 cities. 10 baby care center * 6 cities = 60. The result is an even number

Multiplying Odd Numbers
When two odd numbers are multiplied, the result is always an odd number. The formula is Odd * Odd = Odd. For example: I have to buy 7 play toys for 7 kids. 7 * 7 = 49. The result is an odd number.
When an odd number is multiplied by an even number, the result is always even. The formula is Odd * Even = Even. For example: She has to buy 7 dresses for 4 kids. 7 * 4 = 32. The result is an even number.
These are the multiplication rules for even and odd numbers.

Outer product of vectors in R^3

Outer product of vectors in R^3
Cross product definition:
The outer product is also called the cross product of vectors. The formal definition is as follows:
If vector x = (x1,x2,x3) and vector y = (y1,y2,y3) are vectors in R^3, then their outer product is denoted by x X y and is defined as
x X y = (x1,x2,x3) X (y1,y2,y3)
         = (x2y3 – x3y2, -(x1y3 – x3y1), x1y2 – x2y1) <- cross="" is="" p="" product="" the="" this="" vector.="">Note that the cross product of two vectors is always another vector. Alternatively in matrix notation we can write the cross product identities as follows:
x X y = (|(x2&x3@y2&y3)|, - |(x1&x3@y1&73)|, |(x1&x2@y1&y2)|)

Properties of cross product:
(1) x X y = -y X x
That is because when we do y X x we interchange the rows of the cross product determinants. Interchange of rows results in negative value of the same determinant.

(2) x X x = 0
That is because if a determinant has two identical rows, then the value of the determinant is zero. When finding x X x, the two rows would be identical.

(3) x X ky = kx X y = k(x X y).
Here, k is a scalar. If one of the rows of a determinant has a common factor k, then the k can be taken out of the determinant and the resulting determinant value when multiplied by k gives us the value of the original determinant.

(4) x X (y + z) = x X y + x X z
This is just distributive property. The distributive property is applicable to determinants there for it also applies to the outer product.

Cross product rules:
The following rules apply to all outer products of vectors. The outer product is also called vector product of two vectors.

(1) The vector product of two vectors is always another vector.
(2) The vector product is not commutative as we saw in the first property above.
(3) The vector product is not defined for vectors in R or R^2. It is defined only for vectors in R^3 or higher order dimensions.

Sample problem:
Find the outer product of the vectors x = (1,2,3) and y = (-1,3,5).
Solution:
x X y = (|(2&3@3&5)|, - |(1&3@-1&5)|, |(1&2@-1&3)|) = (10-9, -(5+3), (3+2)) = (1,-8,5) <- answer.="" p="">

Cost Price

Price can be classified into cost price and selling price. Cost price is the value that is paid by someone to purchase goods or products. Cost price is popularly represented as C.P. Selling price on the other hand is the price that a store or product owner sells a product. Selling price is popularly represented as S.P. Below is two examples on cost price and selling price.
• Example 1: Maria bought action toys for 80 rupees per piece and sold it out for 100 rupees per piece. Here, Rs.80 is the cost price of action toys and Rs.100 is the selling price of the same.
• Example 2: Rohit bought building toys collection for Rs.2500 from online. He sold the same collection for Rs. 2000. Here, Rs. 2500 is the cost price and Rs. 2000 is the selling price of the building toys collection.
Cost price can be classified into four main forms – Actual cost, Last cost, Average cost and Net realizable value.
• Actual Cost Price: Actual cost is the complete amount of a product including duty taxes, service taxes and so on. For example: Christy bought role play toys online for Rs.599 exclusive of shipping charges that is Rs.99. Here, actual cost of the role play toys is 599+99 = Rs.698.
• Last Cost Price: Last cost is the value of an item on its last purchase. For example: I bought by mobile phone at Rs.13500 and now the same phone model’s price is Rs. 10000 only. Here, Rs.13500 is the last cost price of the mobile phone.
• Average Cost Price: Average cost is the average of new and old stock values. For example: the new price of a product is Rs. 10 and the old price was Rs.8 and therefore, the average cost is Rs.9.
• Net Realizable Value: Net realizable value is the average price of a product in the market.  For example: The average value of Barbie dolls in market is Rs.199 and it can also be referred as Net realizable value of Barbie dolls.
These are some basics on Cost Price.

Rational expression

A rational expression (in abbreviation as RE) is one of the forms of expressions in algebra. The rational expression definition is an expression which is in fractional form of two expressions. A rational-expression is generally denoted as p/q, q =! 0.. If ‘p’ and ‘q’ are numbers, then it is a fractional number. If ‘p’ is an expression with variable/s and ‘q’ is just a number, then the RE becomes as a simple expression with each coefficients of p is divided by ‘q’.

Rational-expressions have certain constraints which a normal expression may not. As the first one, we said the denominator expression cannot be zero. Thus, a rational expression can have domain and range restrictions. That is, a RE is not defined for the real zeroes of the denominator part. Thus, the domain has to be excluded for such values of the variable/s, and correspondingly the range is affected.

As a convention, the denominators of REs cannot be left with radicals, negative exponents or complex numbers. Putting back them in proper way is called as solving rational expressions. Let us discuss how to solve rational expressions in such cases.

In case of a RE with a radical term in the form (a + sqrt b) in the denominator, multiply both numerator and denominator by the conjugate of (a + sqrt b), which is (a - sqrt b). Now as per the ‘sum and difference product formula’, the denominator becomes as a2 – b, free from radical terms.

In case of complex numbers, the method is exactly same. This process is also called as ‘rationalizing the denominator’.
REs have more prominent place in rational functions. A rational function f(x) is normally expressed in the form f(x) = [g(x)/h(x)]. The domain, range, continuity, asymptotes are all dependent on the nature of g(x) and h(x). A rational function will have vertical asymptotes at the zeroes of h(x). Determining the horizontal asymptotes (HA) of a rational function is a bit lengthy but can me summarized as below. If the degree of g(x) is ‘m’ and that of h(x) is ‘n’, then,

If, (m – n) < 1, then the HA is y = 0, that is, the x-axis.
If, (m – n) = 0, then the HA is y = b, where ‘b’ is the ratio of leading coefficients.
If, (m – n) = 1, then there is no HA but the slant asymptote, is y = mx + b, where ‘mx + b’ is the quotient part of the long division of g(x)/h(x).
If,(m – n) > 1, neither HA or slant asymptote.

Relatively prime numbers

Mathematics is the study of numbers. In branch of mathematics, we deal with different types of numbers which are grouped together.
 For example: - Odd numbers, even numbers, prime numbers, composite numbers, etc.

A prime number is one which has factors one and itself.
For example: - 2 is a prime number as it has two factors only that is 1 and itself. Similarly numbers like 3, 5, 7, and 11 are also its examples.

Now we talk about Relatively Prime Numbers. What is Relatively Prime - Relatively Prime Definition says they are those numbers which have the greatest common divisor as one only.

They are also called co prime-numbers. They are those numbers which share no common factors except 1. Let us look at some examples to understand this concept better.

How about number 7 and 15? If we make factors of 7, we see that number 7 is divisible by 1 and 7 only. For number 15, we see that it is divisible by 1, 3, 5 and 15. If we look at the factors of both the numbers, we see that the common factors between them are 1 only.

Hence they can be termed as relatively-prime-numbers. Similarly if we have numbers 8 and 22, we see that number 8 has factors 1, 2, 4 and 8 and number 22 has factors 1, 2 and 11.

If we look at the factors of both the numbers, we see that apart from number 1, 8 and 22 has a common factor 2 as well. Therefore we cannot call them as relatively-prime-numbers.

If we look at the Relatively Prime Numbers List from 1 to 10, we can make groups like 2 and 3, 2 and 5, 2 and 7, 2 and 9, 3 and 4, 3 and 5, 3 and 7, 3 and 8, 3 and 10, 4 and 5, 4 and 7, 4 and 9, 5 and 6, 5 and 7, 5 and 8, 5 and 9, 6 and 7, 7 and 8, 7 and 9, 7 and 10, 8 and 9 and 9 and 10.
They are forms a group of Relatively Prime nos. till 10.

We cannot include 2 and 4 in that group as they have two common factors. Similarly number 6 and 10 are also not in that group as they are divisible by 1 and 2.

Dividing Decimal

Steps for dividing decimals. Let us say we have 31.773 ÷ 5.1. Let us set this up by using long division. We have 31.773 on the inside, being divided by 5.1. One this we have to remember that we have to get rid of the decimal in the outside number. Here we will be dividing decimals with decimals So, in math dividing decimals, we have outside number of 5.1, here we need to get rid of this decimal in the outside number.

Basically we have to push this decimal point as to the right as possible. In this particular example we can only push it over one space to the right. And whatever we do to the outside number same is applicable to the inside number as well. Let us learn how to divide decimals with decimals. So since we moved our outside decimal number on space to the right, we have to do the same thing to the inside decimal. Our 5.1 becomes 51 and our 31.773 becomes 317.73. So 31.773 divided by 5.1 is exactly the same as 317.73 divided by 51.

Now because we have got the rid of the decimal in the outside number, we can just divide this using normal long division. 51 goes into 3 in 0 times, as 51 is bigger than 3, moving to the next digit. 51 goes into 31, in zeros times, again since 51 is bigger. Moving on to the next digit, 51 go into 317 how many times? Well we know that 50 goes in 300, so we can say that 51 goes into 317 about 6 times. Multiplying 6 times 1 gives 6, and 6 times 5 gives is 30. Now subtracting the two numbers we get 11, now bringing down the 7. So it is 117 now, 51 goes into 117 about 2 times. Two times one is 2, and 2 times 5 is 10. So we get 102.

Subtracting 102 from 117 remains with 15. Bringing down 3 gives us 153. Before we go further to solve this problem, we would like to place the decimal point. The decimal point moves straight up, that goes directly between the two numbers that is, 6.2. so now how many time does 51 go into 153? So we know that 50 go 3 times to get 150, so 51 may also similar ways. So 3 times 1 is 3 and 3 times 5 is 15, So 51 times 3 gives us 153.and subtracting 153 from 153 , our remainder becomes 0.  This means that we can stop, so we did took the help with dividing decimals by shifting the decimal point and made division simple and easy.