HCF and LCM
Prime Number : A prime number is a natural The H.C.F of two or more than two numbers is
number greater than 1 that has no positive the greatest numbers which divide each of them
divisors other than 1 and itself. For example, 2, without any remainder. Highest Common
3, 5, 7, 11, 13, etc. are prime numbers.
Factor (H.C.F.) or Greatest Common Divisor
Co-Prime Number : Two numbers are said to
be relatively prime, mutually prime, or co- (G.C.D.) or Greatest Common Measure
prime to each other, when they have no (G.C.M.) are synonymous terms.
common factor or the only common positive Methods of finding the H.C.F. of a given set of
factor of the two numbers, is 1. In other words, numbers:
two numbers are said to be Co-prime if their Method I : Prime Factorization method :
H.C.F. is 1. (COMMON TERMS WITH LEAST
Factors : The numbers are said to be factors of POWER)
a given number when they exactly divide that Express each one of the given numbers as the
number. Thus, factors of 18 are 1, 2, 3, 6, 9 and product of prime factors. The product of least
18.
powers/index of common prime factors gives
Common Factors : A common factor of two or
more numbers is a number which divides each H.C.F.
of them exactly. Method II : Successive Division method :
Thus, each of the numbers – 2, 4 and 8 is a Divide the larger number by the smaller one.
common factor of 8 and 24. Now, divide the divisor by the remainder.
Multiple : When a number is exactly divisible Repeat the process of dividing the preceding
by another number, then the former number is number by the remainder last obtained till zero
called the multiple of the latter number. Thus, is obtained as remainder. The last divisor is the
45 is a multiple of 1, 3, 5, 9, 15 and 45.
required H.C.F.
Common Multiple : A common multiple of two
or more numbers is a number which is exactly LCM
divisible by each of them. For example, 12, 24 L.C.M. of two or more given numbers is the
and 36 is a common multiple of 3, 4, 6 and 12. smallest number which is divisible by all the
Prime factorization : If a natural number is given numbers.
expressed as the product of prime numbers, Methods of finding the L.C.M. of a given set of
then the factorization of the number is called its numbers:
prime factorization. Prime factorization of a Method I : Prime Factorization method :
natural number can be expressed in the (ALL THE TERMS WITH HIGHEST
exponential form. For example: POWER)
• 24 = 2 x 2 x 2 x 3 = 23 x 3 Express each one of the given numbers as the
• 420 = 2 x 2 x 3 x 5 x 7 = 2² x 3 x 5 x 7 product of prime factors. The product of the
greatest powers/index of common prime factors
HCF
gives L.C.M.
Method II : Division method
1
, HCF and LCM
x = 432
Other important formula related to H.C.F. and
L.C.M. 2. LCM of two numbers is 225 and their
HCF is 5. If one number is 25, the other
1) H.C.F. of given fractions = H.C.F. of number will be:
numerator / L.C.M. of the denominator
2) L.C.M. of given fractions = L.C.M. of Explanation:
numerator / H.C.F. of the denominator
3) Product of two numbers (First number x Product of two numbers (First number x Second
Second Number) = H.C.F. X L.C.M. Number) = H.C.F. X L.C.M.
4) H.C.F. of a given number always divides
its L.C.M. 25 * x = 225 * 5
5) The largest number which divides x, y, z
x = (225 * 5)/25
to leave remainder R in each case =
H.C.F. of (x-R), (y-R), (z-R). x = 45
6) Largest number which divides x, y, z to
leave same remainder = H.C.F. of (y-x), 3. The L.C.M. of two numbers is 1820 and
(z-y), (z-x). their H.C.F. is 26. If one number is 130 then
7) Largest number which divides x, y, z to
the other number is :
leave remainder a,b,c = H.C.F. of (x-a),
(y-b), (z-c). Explanation:
8) Least number which when divided by x,
y, z and leaves a remainder R in each L.C.M. of two numbers = 1820
case = (L.C.M. of x, y, z) + R.
H.C.F. of those numbers = 26
1. The LCM of two numbers is 864 and their One of the number is 130
HCF is 144. If one of the number is 288, the
Product of two numbers (First number x Second
other number is :
Number) = H.C.F. X L.C.M.
Explanation:
130 * x = 1820 * 26
Product of two numbers (First number x Second
x = (1820 * 26)/130
Number) = H.C.F. X L.C.M.
x = 364
288 * x = 864 * 144
4. The HCF of two numbers 12906 and 14818
x = (864 * 144)/288
is 478. Find their LCM.
2
Prime Number : A prime number is a natural The H.C.F of two or more than two numbers is
number greater than 1 that has no positive the greatest numbers which divide each of them
divisors other than 1 and itself. For example, 2, without any remainder. Highest Common
3, 5, 7, 11, 13, etc. are prime numbers.
Factor (H.C.F.) or Greatest Common Divisor
Co-Prime Number : Two numbers are said to
be relatively prime, mutually prime, or co- (G.C.D.) or Greatest Common Measure
prime to each other, when they have no (G.C.M.) are synonymous terms.
common factor or the only common positive Methods of finding the H.C.F. of a given set of
factor of the two numbers, is 1. In other words, numbers:
two numbers are said to be Co-prime if their Method I : Prime Factorization method :
H.C.F. is 1. (COMMON TERMS WITH LEAST
Factors : The numbers are said to be factors of POWER)
a given number when they exactly divide that Express each one of the given numbers as the
number. Thus, factors of 18 are 1, 2, 3, 6, 9 and product of prime factors. The product of least
18.
powers/index of common prime factors gives
Common Factors : A common factor of two or
more numbers is a number which divides each H.C.F.
of them exactly. Method II : Successive Division method :
Thus, each of the numbers – 2, 4 and 8 is a Divide the larger number by the smaller one.
common factor of 8 and 24. Now, divide the divisor by the remainder.
Multiple : When a number is exactly divisible Repeat the process of dividing the preceding
by another number, then the former number is number by the remainder last obtained till zero
called the multiple of the latter number. Thus, is obtained as remainder. The last divisor is the
45 is a multiple of 1, 3, 5, 9, 15 and 45.
required H.C.F.
Common Multiple : A common multiple of two
or more numbers is a number which is exactly LCM
divisible by each of them. For example, 12, 24 L.C.M. of two or more given numbers is the
and 36 is a common multiple of 3, 4, 6 and 12. smallest number which is divisible by all the
Prime factorization : If a natural number is given numbers.
expressed as the product of prime numbers, Methods of finding the L.C.M. of a given set of
then the factorization of the number is called its numbers:
prime factorization. Prime factorization of a Method I : Prime Factorization method :
natural number can be expressed in the (ALL THE TERMS WITH HIGHEST
exponential form. For example: POWER)
• 24 = 2 x 2 x 2 x 3 = 23 x 3 Express each one of the given numbers as the
• 420 = 2 x 2 x 3 x 5 x 7 = 2² x 3 x 5 x 7 product of prime factors. The product of the
greatest powers/index of common prime factors
HCF
gives L.C.M.
Method II : Division method
1
, HCF and LCM
x = 432
Other important formula related to H.C.F. and
L.C.M. 2. LCM of two numbers is 225 and their
HCF is 5. If one number is 25, the other
1) H.C.F. of given fractions = H.C.F. of number will be:
numerator / L.C.M. of the denominator
2) L.C.M. of given fractions = L.C.M. of Explanation:
numerator / H.C.F. of the denominator
3) Product of two numbers (First number x Product of two numbers (First number x Second
Second Number) = H.C.F. X L.C.M. Number) = H.C.F. X L.C.M.
4) H.C.F. of a given number always divides
its L.C.M. 25 * x = 225 * 5
5) The largest number which divides x, y, z
x = (225 * 5)/25
to leave remainder R in each case =
H.C.F. of (x-R), (y-R), (z-R). x = 45
6) Largest number which divides x, y, z to
leave same remainder = H.C.F. of (y-x), 3. The L.C.M. of two numbers is 1820 and
(z-y), (z-x). their H.C.F. is 26. If one number is 130 then
7) Largest number which divides x, y, z to
the other number is :
leave remainder a,b,c = H.C.F. of (x-a),
(y-b), (z-c). Explanation:
8) Least number which when divided by x,
y, z and leaves a remainder R in each L.C.M. of two numbers = 1820
case = (L.C.M. of x, y, z) + R.
H.C.F. of those numbers = 26
1. The LCM of two numbers is 864 and their One of the number is 130
HCF is 144. If one of the number is 288, the
Product of two numbers (First number x Second
other number is :
Number) = H.C.F. X L.C.M.
Explanation:
130 * x = 1820 * 26
Product of two numbers (First number x Second
x = (1820 * 26)/130
Number) = H.C.F. X L.C.M.
x = 364
288 * x = 864 * 144
4. The HCF of two numbers 12906 and 14818
x = (864 * 144)/288
is 478. Find their LCM.
2
