Download Quantitative Aptitude HCF and LCM Tutorial

Quantitative Aptitude HCF and LCM Tutorial
(Study Material)
Prime factorisation
If a natural number is expressed as the product of prime numbers, then the factorisation
of the number is called its prime (or complete) factorisation.
A prime factorisation of a natural number can be expressed in the exponential form.
For example:
(i) 48 = 2×2×2×2×3 = 24×3
(ii) 420 = 2×2×3×5×7 = 2² ×3×5×7.
Least Common Multiple (abbreviated L.C.M.) of two natural numbers is the smallest
natural number which is a multiple of both the numbers.
Highest Common Factor (abbreviated H.C.F.) of two natural numbers is the largest
common factor (or divisor) of the given natural numbers. In other words, H.C.F. is the
greatest element of the set of common factors of the given numbers.
H.C.F. is also called Greatest Common Divisor (abbreviated G.C.D.)
Co-prime numbers: Two natural numbers are called co-prime numbers if they have
no common factor other than 1.
In other words, two natural numbers are co-prime if their H.C.F. is 1.
Some examples of co-prime numbers are: 4, 9; 8, 21; 27, 50.
Relation between L.C.M. and H.C.F. of two natural numbers
The product of L.C.M. and H.C.F. of two natural numbers = the product of the numbers.
Note. In particular, if two natural numbers are co-prime then their L.C.M. = the product
of the numbers.
Factors
Telangana History in Telugu SSC CGL Previous Papers TSPSC Group II Study Material
In a division, if a number / divides a number M completely (exactly) or in other words, if
M is exactly divisible by f then f is the factor of M.
Example: 5 divides 35 completely, so. 5 is a factor of 35.
Similarly. 2. 3, 4, 6 are all factors of 12. because each of the numbers 2, 3. 4. and 6 will
divide 12 completely or, in other words 12 is divisible by 2. 3. 4 and 6.
Multiples
From the above concept, if f is a factor of M, then M is a multiple of f.
Example: 63 is completely divisible by 7, 3, 9, 21. So. 63 is a multiple of 7 or 3 or 9 or 21.
Principle of Prime Factorisation
Any natural number (>1) is cither prime or non-prime (composite)
The principle of prime factorisation states:
Each non-prime (composite) number can be uniquely broken (reduced) into two or
more prime numbers (prime factors). In other words, each non-prime number is
divisible by any of the prime numbers.
With the use of this principle, a non-prime number is broken into its prime factor by
dividing it with different prime numbers. This is known as division method of
factorisation of a number. The same is explained in the following example.
Thus, 20570 = 2 x5 x 11 x 11 x 17.
Hence, if the number is even, the division should start with 2; otherwise, rest of the
prime numbers should be tried in succession.
Highest Common Factor (HCF)
Telangana History in Telugu SSC CGL Previous Papers TSPSC Group II Study Material
If two or more numbers are broken into their prime factors (as explained in 2.3), then
the product of the maximum common prime factors in the given numbers is the H.C.F.
of the numbers.
In other words, the HCF of two or more numbers is the greatest number (divisor) that
divides all the given numbers exactly. So. HCF is also called the Greatest Common
Divisor (GCD).
Example: Find the HCF of 72, 60. 96.
LCM (Lowest Common Multiple)
The LCM of two or more than two numbers is the product of the highest powers of all
the prime factors that occur in these numbers.
Example: Find the LCM of 36, 48, 64 and 72
Telangana History in Telugu SSC CGL Previous Papers TSPSC Group II Study Material
Product of Two Numbers
HCF of numbers x LCM of numbers = Product of numbers.
i.e., if the numbers are A and B. then
HCF of A and B x LCM of A and B = A x B
Difference Between HCF and LCM
LCM of x, y and z
HCF of x, y and z
is the Highest Divisor which can exactly divide is the Least Dividend which is exactly
x, y and z.
divisible by .r, y and z.
Rapid Information List
Type of Problem
Approach to Problem
Telangana History in Telugu SSC CGL Previous Papers TSPSC Group II Study Material
Find the GREATEST NUMBER that will
exactly divide y and z.
Required number = HCF of x. y and z (greatest
divisor)
Find the GREATEST NUMBER that will
divide .t, y and z leaving remainders a. b
and c respectively.
Required number (greatest divisor)
= HCF of (x - a), (y - b) and (z - c)
Find the Least Number which is exactly
divisible by x, y and z.
Required number = LCM of x, y and z (least
dividend)
Then, it is always observed that (x - a) =(y - b) =
Find the LEAST NUMBER which when
(z - c) = K (say). ./Required number
divided by x, y and z leaves the remainders
a. b and c respectively.
= {LCM of x, y and z) - (k)
Find the LEAST NUMBER which when
divided by x. y and z leaves the same
remainder *r* each case.
Required number = (LCM of x, y and z) + r.
Required number
Find the GREATEST NUMBER that will
divide a. y and z leaving the same remainder
in each case.
= HCF of (.r - y). (v - z) and (z - x)
LCM of x. y and z = L
Find the n-digil GREATEST NUMBER
which when divided by x. y and z.
(a) leaves no remainder (i.e.. exactly
divisible)
(b) leaves remainder K in each case.
n-digit greatest number/L = R
(Step I)
(Step 2)
By Rule 1 (Chapter 1).
(a)
-R
Required number = n-digit greatest number
(b) Required number
.
= [n-digit greatest number - R] + K
LCM of x. y and z = L
Find the n-digit SMALLEST NUMBER
which when divided by .t, v ;ind c
(a) leaves no remainder (i.e., exactly
divisible)
(b) leaves remainder K in each case.
digit greatest number /L
remainder = R
(a)
(Step 1)
(Step 2)
By Rule II (Chapter I)
Required number
=n-digit smallest number + (L - R)
(b) Required number
Telangana History in Telugu SSC CGL Previous Papers TSPSC Group II Study Material
= n-digit smallest number + (L - R) + K
HCF of fractions
Find the HCF of x/a , a/b and m/n
HCF of numerators / LCM of denominators
LCM of fractions
Find the LCM of x/a , a/b and m/n
l.CMof numerators / HCF of denominators
Step 1 Find the HCF of the given numbers
without decimal.
Find the HCF of decimal numbers
Step 2 Put the decimal point (in the HCF of Step
1) from right to left according to the MAXIMUM
decimal places among the given numbers.
Step l Fiiid the LCM of the given numbers
without decimal.
Find the LCM of decimal numbers



Telangana History in Telugu
Quantitative Aptitude
Reasoning Study Material PDF



SSC CGL Previous Papers
TSPSC Group II Study Material
Study Material
Step 2 Put the decimal point (in the LCM of Step
1) from right to left at the place equal to the 1
MINIMUM decimal places among the given
numbers.
Telangana History in Telugu SSC CGL Previous Papers TSPSC Group II Study Material