1. Factor and Multiples :
Factoring is like taking a number apart. It means to express a number as the product of its factors. Factors are either composite numbers or prime numbers (except that 0 and 1 are neither prime nor composite). The number 12 is a multiple of 3, because it can be divided evenly by 3. 3 x 4 = 12.
2. Highest common factor (H.C.F) or Greatest Common Measure (G. C. M) or Greatest Common Divisor (G. C. D) :
The least number which is exactly divisible by each of the given numbers is called the least common multiple of those numbers. The H.C.F of two or more then two numbers is the greatest number that divides each of them exactly.
There are two methods of finding the H.C.F of a given set of numbers :
1. Factorization Method : Express each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives H.C.F.
2. Division Method : Suppose we have to find the H.C.F. of two given numbers. Divide the larger number by the smaller one. Now, divide the divisor by the remainder Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The divisor is the required H.C.F.
Steps 1: Find all the factors of each given number.
Step 2: Find common factors of the given number.
Step 3: The greatest of all the factors obtained in Step 2, is the required highest common factor (H.C.F).
Finding the H.C.F of more than two numbers : Suppose we have to find the H.C.F. of three numbers. Then, H.C.F. of three given numbers.
Similarly, the H.C.F. of more than given numbers may be obtained.
3. Last Common Multiple (L.C.M) :
The least number which is exactly divisible by each one of the given numbers is called their L.C.M.
1. Factorization Method of Finding L.C.M : Resolve each of the given numbers into a product of prime factors. Then, L.C.M. is the product of highest powers of all the factors.
2. Common Division Method (Short-cut Method) of Finding L.C.M : Arrange the given numbers in a row in any order. Divide by a number which divides exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required L.C.M OF the given numbers.
4. Product of two numbers = Product of their H.C.F and L.C.M.
5. C-primes : Two numbers are said to be co-primes if their H.C.M is 1.
6. H.C.F and L.C.M. of Fraction :
1. H.C.F. = H.C.F.of Numerators / L.C.M.of Denominators
2. L.C.M = L.C.M of Numerators / H.C.F.of Denominators
7. H.C.F. and L.C.M of Decimal Fractions :
In given numbers, make the same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find H.C.F or L.C.M as the case may be. Now. In the result, mark off as many decimal places as are there in each of the given numbers.
Steps to solve H.C.F. and L.C.M. of decimals:
Step 1: Convert each of the decimals to like decimals.
Step 2: Remove the decimal point and find the highest common factor and least common multiple as usual.
Step 3: In the answer (highest common factor /least common multiple), put the decimal point as there are a number of decimal places in the like decimals.
8. Comparison of Fractions :
Find the L.C.M of the denominators of the given fractions, Convert each of the fractions into an equivalent fraction with L.C.M as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.
There are two main ways to compare fractions:
1. Using decimals,
2. Using the same denominator.