If a number divides a larger number, it is said to be a factor of the larger number. Hence, 3 would be a factor of 6. The factors of 28 are 1, 2, 7, 14, as each of these divides the number 28.
Number of factors
If a number can be expressed as C = am × bn × cp × … then the number of factors that the number has, is: (m + 1)(n + 1)(p + 1).
Any number, when multiplied with another number, yields a multiple. A number may thus have numerous multiples. For instance, 6, 9, 12, 15, 18, … are multiples of 3 as 3 multiplied with other numbers yields its multiples.
We may be asked to find multiples of a number less than another number.
Example: How many multiples of 6 are less than 350?
Solution: To find the number of multiples of 6 less than 350, we divide 350 by 6 to get 58. Hence there will be 58 multiples of 6 less than 350.
Example: How many distinct factors exist of 12?
Solution: The factors of 12 are 22 × 3.
Using the above formula, we get number of factors = (2 + 1)(1 + 1) = 6.
In this case, we can check this and see that the factors are: 1, 2, 3, 4, 6, 12, or 6 factors.
Highest Common Factor - HCF
A number, which is a factor of two or more numbers, is called a common factor.
HCF — The Highest Common Factor or Greatest Common Divisor (GCD) or Greatest Common Measure (GCM) of two or more numbers is the greatest number that divides each one of them exactly.
HCF is found by expressing each one of the given numbers as the product of prime factors. Then the common factors are taken out and the product of these factors is the HCF.
To find HCF
By factorization: Find the factors of the given numbers. Now, choose common factors and take the product of these factors.
Example: Find the HCF of 6, 8 and 10
Solution: Factors of 6 = 2 × 3 Factors of 8 = 2 × 2 × 2 Factors of 10 = 2 × 5
Note that the only common factor that occurs in all the three numbers is 2. 3 and 5 occur only in one number each. Hence the HCF is 2.
HCF By Division Method: To find the HCF of two given numbers, divide the larger number by the smaller. Repeat the process of dividing the preceding divisor by the remainder last obtained, till a remainder zero is obtained. The last divisor is the HCF of given numbers.
- Divide the larger number by the smaller
- Divide the previous divisor by the previous remainder.
Continue the process till the remainder of 0 is obtained. The divisor, when the remainder is 0, is the HCF.
Example: Find the HCF of 24 and 34.
The remainder at the last stage is 0, and the divisor at this stage (2) is the HCF.
By factorization method:
Factors of 24 = 23 × 3
Factors of 34 = 2 × 17
The common factor in the 2 numbers is 2.
Hence, for the purpose of the exam, the student should prefer the factorisation method as it is faster.
Also, since choices are given, the student needs to develop a judgement as to what is the HCF of numbers rather than finding it out.
To find the HCF of three numbers: take the HCF of (HCF of any two and the third number). Similarly, the HCF of more than three numbers may be obtained.
Lowest Common Multiple — LCM
The least number which is exactly divisible by each one of the given numbers, is called their LCM.
LCM is found by resolving the given numbers into their prime factors and then finding the product of the highest powers of all the factors that occur in the given numbers. The product will be the LCM.
To find LCM:
Factorise the given numbers. Choose the highest powers of each prime factor and multiply them together.
Example: Find the LCM of 15, 18, and 30.
Solution: Factorising the numbers:
15 = 3 × 5, 18 = 2 × 32, 30 = 2 × 3 × 5.
Choosing the highest powers, we get 2 × 32 × 5 = 90.
LCM of three numbers = LCM of (LCM of any two & third). Similarly, the LCM of more than three numbers can be obtained.
For instance, to find the LCM of 15, 18, and 30, we can also do as follows: LCM of 15 and 30 = 30.
LCM of 30 and 18 = 90
Hence LCM of three numbers can also be found out by taking two at a time.
If a single remainder is given, then first the LCM of those numbers is calculated and subsequently, that single remainder is added in that.
If for different numbers different remainders are given then the difference between the numbers and their respective remainders would be equal. First, the LCM of the numbers given is calculated and then the common difference between the numbers and their respective remainders is subtracted from that.
The product of two numbers M and N would always be equal to the product of their HCF & LCM i.e.
Product of Two Numbers = (Their H.C.F.) × (Their L.C.M.).
LCM of Fractions = LCM of Numerators ÷ HCF of Denominators
HCF of Fractions = HCF of Numerators ÷ LCM of Denominators
Example: Find the HCF and LCM of 2/3 and 7/12.
Solution: HCF = HCF of Numerators ÷ LCM of Denominators = HCF (2, 7)/LCM (3, 12) = 1/12. LCM = LCM of Numerators ÷ HCF of Denominators = LCM (2, 7)/HCF (3, 12) = 14/3.
Note that the product of the two fractions is always equal to the product of LCM and HCF of the two fractions. The product of the two fractions = 2/3 × 7/12 = 7/18.
The product of the LCM and HCF = 1/12 × 14/3 = 7/18.
To find HCF & LCM of decimal fractions, consider these numbers without a decimal point and find HCF or LCM, as the case may be. In a result, mark off as many decimal places as are there in each of the given numbers.
Example: Find the HCF and LCM of 0.12 and 0.3.
Solution: Consider the terms without the decimal: they become 12 and 30.
HCF of 12 and 30 = 3, hence HCF of the given numbers is 0.03 (putting two decimals).
LCM of 12 and 30 = 60, hence LCM of given numbers is 0.60.