Factors and multiple : If a number a divides number b exactly, we say that a is a factor of b in this case , b in called a multiple of a .
Highest common factor (H.C.F) or greatest common measure (G. C.M.) or greatest common Divisor (G.C.D.). The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly.
There are two methods of finding the HCF of a given set of numbers :
1. Factorizing method : Express each one of the given number as the product of prime factors. The product of least of common prime factors gives HCF.
Example: Lets say we have to find the HCF of 24 and 54.
Lets prime factorized the number 24 and 54.
24 = 2 * 2 * 2 * 3
54 = 2 * 3 * 3 * 3
only 2 *3 common in both the numbers.
Therefore, 6 is the HCF of 24 and 54.
2.Long Division Method: Suppose we have to find the HCF of two given numbers. Divide the larger number by the smaller opne. Now divide the larger number by the smaller number 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 last divisor is required HCF.
Taking above numbers 24 and 54.
Dividing 54 by 24 we will get 2 as quotient and 6 as remainder.
Now divide the divisor 24 by the remainder 6. We will get 0 remainder. Since the number 6 was the last divisor, we have 6 as the H.C.F.
Finding the HCF of more than two number: Suppose we have to find the HCF of three numbers. Then, HCF, of { HCF of any two and the third number} gives the HCF of three given numbers.
Similarly, the HCF of more than three number is obtained.
Least Common Multiple( LCM): The least number which is exactly divisible each one of the given numbers is called their LCM.
1 Factorization method of Finding L.C.M : Resolve each one of the given numbers in to a P roduct of Prime factors then , L .C. M is the Product of highest Product 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 divisible . Repeat the above Process til no 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. Co-primes : Two numbers are said to be co-primes if their H.C.F. is 1.
6. H.C.F and L.C .M of Fractions :
HCF= ( HCF . of Numbers / LCM. of Denominators ) 2. LCM.= (of Numbers | HCF of Denominators
7. HCF and LCM 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 HCF or LCM as the case may be Now , in the result, mark off many decimal Places as are there in each of the given numbers,
8,Comparison of Fractions : Find the LCM of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with LCM as the denominator , by multiplying both the numerator and denominator by the same number the resultant fraction with the greatest numerator is the greatest
Solved Questions:
Q1. Find the Least Number which when divided by 12,18,36 and 45 leaves the remainder 8,14,32 and 41 respectively.
(a) 180 (b) 178
(c) 186 (d) 176
Answer: (d) 176
Solution:
First we have to find the relationship between the Divisors and the remainder.
We see that 12-8=4 , 18-14 = 4, 36-32=4 and 45-41 = 4
The rule here is to find the LCM of 12,18,36 and 45. That is 180.
Now, the required number is obtained by subtracting the common difference (4) obtained earlier from the LCM of the number.
i.e. 180 - 4 = 176
2. The ratio of the two numbers is 3:4 and their LCM is 180. The second number is
(a) 45 (b) 90
(c) 30 (d) 60
Answer: (d) 60
Solution:
The point to understand here is that,
when we find ratio of two numbers then the HCF of the numbers is eliminated while reducing the ratio to the lowest term. Thus, the two numbers are 3 x HCF and 4 x HCF
Now, Let the HCF be x. Then using the formula
LCM x HCF = First Number and Second Number
180 x HCF = (3 x HCF)(4 x HCF)
=> 180/(3x4) = HCF
=> 15 = HCF
Therefore, the second number is 4 x15 = 60
3. A farmer has 945 cows and 2475 sheep. He farms then into flocks, keeping cows and sheep separate and having the same number of animals in each flock. If these flocks are as large as possible, then the maximum number of animals in each flock and total number of flocks required for the purpose are respectively
(a) 15 and 228 (b) 9 and 380
(c) 45 and 76 (d) 46 and 75
Answer:(c) 45 and 76
Solution:
Here we just have to find the Highest Common Factor of these two numbers (945,2475). It will come out to be 45. Therefore, looking at the options the CORRECT answer is option (c).
4. If x:y be the ratio of two whole numbers and z be their h.c.f. then the l.c.m. of those two numbers is
(a) yz (b) xz/y
(c) xy/z (d) xyz
Answer:(d) xyz
Solution:
The main point to note here is that when we find ratio of two numbers then the what is reduced from them is the h.c.f. of both the numbers.
Thus the two numbers are (x.z ) and (y.z)
Again, using the formula,
multiplication of two numbers = (l.c.m.)(h.c.f)
=> (x.z)(y.z) = (z)(l.c.m.)
=> l.c.m. = xyz
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