LOWEST COMMON MULTIPLE

L.C.M. stands for Lowest Common.

LCM

So let us first discuss about what are multiples of the given number.

Multiple: The number obtained by multiplying the given number is called the multiple of given number.

For example: 2 × 1 = 2, 2 × 2 = 4, 2 × 3 = 6, etc. therefore 2, 4, 6, 8, etc. are the multiples of 2.

Lowest Common Multiple: The L.C.M. of two or more given numbers is the lowest (smallest) number that is a multiple of each of the given numbers.

Thus, it is the smallest number which is exactly divisible by each of the given numbers.

For example: L.C.M. of 4 and 6 is 12 as 12 is the least multiple common multiple of 4 as well as 6.

Methods of finding L.C.M.: The following three methods are most commonly used to find the L.C.M.

1) Common Multiple Method:

Step 1: Find a few multiples of each given number.

Step 2: From the multiples obtained in Step 1, select the common ones.

Step 3: The lowest number (multiple) obtained in Step 2 is the required lowest common multiple of the given numbers.

For example: Find L.C.M. of 4, 5 and 10.

Step 1: Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, ......

Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45 ,50, 55, .....

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

Step 2: Common multiples of 4, 5 and 10 = 20, 40, 60, 80, .....

Step 3: smallest common multiple obtained in step 2 is 20.

So, L.C.M. of 4, 5 and 10 is 20.

2) Prime Factor Method:

Step 1: Express each of the given number as a product of its prime factors and then in index form.

Step 2: Product of all the prime factors obtained in step 1 with highest power of each is the L.C.M. of given numbers.

For example: Find L.C.M. of 18 , 24 and 36.

Step 1: 18 = 2 × 3 × 3 = 2¹ × 3²,

24 = 2 × 2 × 2 × 3 = 2³ × 3¹,

36 = 2 × 2 × 3 × 3 = 2² × 3²

Step 2: Since, the prime factors 2 and 3, obtained above, with highest power are 2³ and 3².

Therefore, L.C.M. = 2³ × 3² = 2 × 2 × 2 × 3 × 3 = 72.

3) Common Division Method:

Step 1: Write all the given numbers in a horizontal line, separating them by commas.

Step 2: Divide by a suitable number, that exactly divides at least two of the given numbers. And, write down the quotients and the undivided numbers obtained, below the first line.

Step 3: Repeat the process until we get a line of numbers that are prime to one-another.

Step 4: The product of all the divisors and the numbers obtained in the last line will be the required L.C.M.

For example: Find L.C.M. of 16, 20 and 24.

16, 20, 24

2 16, 20, 24

8, 10, 12

2 16, 20, 24

2 8, 10, 12

2 4, 5, 6

2, 5, 3

Therefore, L.C.M = 2 × 2 × 2 × 2 × 3 × 5 = 240.

As we know H.C.F. stands for highest common factor.

HCF

Let us first discuss about factors.

Factors: When two or more natural numbers are multiplied together, the result is referred as their product, and each of the numbers multiplied is called a factor of this product.

For example: product of 3 and 7 is 21, therefore 3 and 7 are factors of 21, product of 2, 3 and 5 is 30, therefore each of 2, 3 and 5 is a factor of 30.

In other words

Any natural number that divides a given number completely is called a factor of the given number.

For example: 5 divides 20 completely → 5 is a factor of 20,

7 divides 42 completely → 7 is a factor of 42.

As each of 1, 2, 3, 4, 6, 8, 12 and 24 divides 24 completely, so factors of 24 are 1, 2, 3, 4, 6, 12 and 24.

Factors of F24 = 1, 2, 3, 4, 6, 12, 24.

1 (one) is a factor of every number.
Every number is a factor of itself.
Zero (0) can not be a factor of any number.

Prime factors: Any prime number that divides completely a given natural number is called a prime factor of the given number.

For example: factors of 24 are 1, 2, 3, 4, 6, 12 and 24. Out of these 2 and 3 are the prime numbers and so the factors 2 and 3 are called the prime factors of 24.

HIGHEST COMMON FACTOR: The H.C.F. of two or more given numbers is the greatest number that divides each of the given numbers.

For example: The greatest number that can divide both 18 and 24 completely is 6; therefore, H.C.F. of 18 and 24 = 6.

METHODS OF FINDING H.C.F.

For finding the H.C.F. of two or more given numbers, any of the following three methods can be used:

1) Common Factor Method:

Step 1: Firstly find all the possible factors of each given number.

Step 2: From the factors obtained in Step 1, select the common factors.

Step 3: Out of the common factors, obtained in Step 2, take the highest factor, which is the Highest Common Factor (H.C.F.) of the given numbers.

For example:

Step 1: Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36 and

Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

Step 2: Factors that are common to 36 and 48 = 1, 2, 3, 4, 6, 12

Step 3: From the result of Step 2, the highest common factor = 12

2) Prime factor method:

Step 1: Split each given number into its prime factors.

Step 2: Select the common prime factors.

Step 3: Multiply the prime factors obtained in Step 2.

The product so obtained is the H.C.F. of the given numbers.

For example: Find H.C.F. of 15 and 25.

Step 1: Prime factors of 15 are 3 and 5, since 5 × 3 = 15, prime factors of 25 are 5 and 5, since 5 × 5 = 25

Step 2: Since the common prime factor is 5 only,

Step 3: Therefore H.C.F. of 15 and 25 is 5.

Any two numbers that do not have a common prime factor are called co-prime numbers. e.g 2 and 3, 5 and 7.

The H.C.F. of two co-prime numbers is always 1.

3) Division method:

Step 1: Divide the greater number by the smaller number.

Step 2: By the remainder of division in Step 1, divide the smaller number.

Step 3: By the remainder in Step 2, divide the remainder obtained in Step 1.

Step 4: Continue in the same way till no remainder is left. The last divisor is the required H.C.F.

For example: Find the H.C.F. of 36 and 60.

Step 1: 60 = 36 × 1 + 24

Step 2: 36 = 24 × 1 + 12

Step 3: 12 = 12 × 2 + 0

Since the last divisor is 12, therefore H.C.F. is 12.

Galois

Évariste Galois was born on October 25, 1811, near Paris. He took his first mathematics course when he was 15 and quickly mastered the work of Legendre and Lagrange. At the age of 18, Galois wrote his important research on the theory of equations and submitted it to the French Academy of Sciences for publication. The paper was given to Cauchy for refereeing. Cauchy impressed by the paper, agreed to present it to the academy, but never did. At the age of 19, Galois Prize in Mathematics, given by the French Academy of Sciences. The paper was given to Fourier, who died shortly thereafter. Galois's paper was never seen again.

"Galois at seventeen was making discovering of epochal significance in the theory of equations, discoveries whose consequences are not yet exhausted after more than a century."

E.T. Bell, Men of Mathematics

Galois stamp

Galois twice failed his entrance examination to 1'École Polytechnique. He did not know some basic mathematics, and he did mathematics almost entirely in his head, to the annoyance of the examiner. Legend has it that Galois became so enraged at the stupidity of the examiner that he threw an eraser at him.

Galois spent most of the last year and half of his life in prison for revolutionary political offenses. While in prison, he attempted suicide, and prophesied his death in a duel. On may 30, 1832, Galois was shot in a duel and died the next day at the age of 20. The life and death of Galois have long been a source of fascination and speculation for mathematics historians. One article [1] argues that three of the most widely read accounts of Galois's life are highly fictitious.

Among the many concepts introduced by Galois are normal subgroups, isomorphism, simple groups, finite fields, and Galois theory. His work provided a method for disposing of several famous constructability problems, such as trisecting an arbitrary angle and doubling a cube. Galois's entire collected work full only 60 pages.

NUMBER SYSTEM

Natural numbers: Counting numbers are called natural number.

Thus, 1, 2, 3, 4, ...... are all natural number.

Whole numbers: All counting numbers, together with 0, form the set of whole numbers.

Thus, 0, 1, 2, 3, 4, ..... are all whole numbers.

Integers: All counting numbers, zero and negatives of counting numbers, form the set of integers.

Thus, ....., -3, -2, -1, 0, 1, 2, 3,...... are all integers.

Set of positive integers = { 1, 2, 3, 4, 5, .......... }

Set of negative integers = { -1, -2, -3, -4, ....... }

Set of non-negative integers = { 0, 1, 2, 3, 4, .......}

Set of non-positive integers = { 0, -1, -2, -2, -4, .......}

Even numbers: All multiples of two are called even numbers.

Since, 2, 4, 6, 8, 10, 12, 14, 16, 18, ...... is a multiple of 2.

Therefore each of 2, 4, 6, 8, 10, 16, 18, is an even number.

Every even number is divisible by 2 or 2 is a factor of every even number.

For example: 2 × 1 = 2, 2 × 2 = 4, 2 × 3 = 6, etc. therefore 2, 4, 6, 8, etc. are the multiples of 2.

Multiples: The number obtained by multiplying the given number is called the multiple of given number.

Odd numbers: Numbers which are not divisible by 2 are called odd numbers.

Since, none of 1, 3, 5, 7, 9, 11, 13, .... is divisible by 2, so each of 1, 3, 5, 7, 9, 11, 13, ...... is an odd number.

Every number is either even or odd.
A number cannot be both even as well as odd.
All prime numbers except 2 are odd.

Prime number: A natural number that is divisible by not one (1) and itself is called a prime number.

For example: 2 is divisible only by 1 and itself, therefore 2 is prime number; 5 is divisible by only 1 and itself, therefore 5 is prime number.

To be more clear, note:

If a natural number has only two factors, it is a prime number.
1 (one) is not a prime number as it has only one factor.
3 is a prime number as it has only two factors, which are 1 and 7.
10 is not a prime number as it has more than two factors i.e.1, 2, 5, 10.

All prime numbers less than 100 are 2, 3, 5, 7, 11, 13 ,17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Composite numbers: Every natural number that has more than two factors is called a composite number.

Since, factors of 10 are 1, 2, 5 and 10.

Therefore, 10 is a composite number.

Similarly, each of 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, etc. is a composite number.

Every even number greater than 2 is a composite number.

Perfect number: A number, the sum of whose factors (except the number itself), is equal to the number, is called a perfect number.

For example: The factors of 6 are 1, 2, 3 and 6. And, 1+ 2+ 3 = 6, so 6 is perfect number.

Co-prime numbers: Any two numbers that do not have a common prime factor are called co-prime numbers.

For example: 2 and 3, 5 and 7 are the examples of co-prime numbers.

Twin prime: The two prime numbers whose difference is 2 are called twin prime numbers.

Important facts:

The smallest prime number is 2.
The only even prime number is 2.
The first odd prime number is 3.
1 is unique number which is neither prime nor composite.
The least composite number is 4.
The least odd composite number is 9.
All natural numbers are whole numbers.
All whole numbers are not natural numbers as 0 is a whole number but not a natural number.