Quantitative Aptitude HCF and LCM Tutorial (Study Material)
Quantitative Aptitude HCF and LCM Tutorial ( Download PDF)
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
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)
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
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
| HCF of x, y and z | LCM of x, y and z |
| is the Highest Divisor which can exactly divide x, y and z. | is the Least Dividend which is exactly divisible by .r, y and z. |
Rapid Information List
| Type of Problem | Approach to Problem |
| 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) |
| Find the LEAST NUMBER which when divided by x, y and z leaves the remainders a. b and c respectively. | Then, it is always observed that (x – a) =(y – b) = (z – c) = K (say). ./Required number
= {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. |
| Find the GREATEST NUMBER that will divide a. y and z leaving the same remainder in each case. | Required number
= HCF of (.r – y). (v – z) and (z – x) |
| 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. |
LCM of x. y and z = L (Step I)
n-digit greatest number/L = R (Step 2) By Rule 1 (Chapter 1). (a) Required number = n-digit greatest number – R (b) Required number . = [n-digit greatest number – R] + K |
| 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. |
LCM of x. y and z = L (Step 1)
digit greatest number /L (Step 2) remainder = R By Rule II (Chapter I) (a) Required number =n-digit smallest number + (L – R) (b) Required number = n-digit smallest number + (L – R) + K |
| Find the HCF of x/a , a/b and m/n | HCF of fractions
HCF of numerators / LCM of denominators |
| Find the LCM of x/a , a/b and m/n | LCM of fractions
l.CMof numerators / HCF of denominators |
| Find the HCF of decimal numbers | Step 1 Find the HCF of the given numbers without decimal.
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. |
| Find the LCM of decimal numbers | Step l Fiiid the LCM of the given numbers without decimal.
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. |