H.C.F. & L.C.M. of Numbers
I.) Factors
When a divides b exactly , then we say that a is a factor of b and we write , a/b .
Example :
1) Prime factorisation of 22475 ?
Solution :
5 | 22475
5 | 4495
29 | 899
31 | 31
| 1
Thus, prime factorisation of
22475 = 5×5×29×31.
II.) Multiplies
When a divides b exactly , then we say that b is a ,multiple of a .
III.) H.C.F. (Highest Common Factor )
or
G.C.D. (Greatest Common Factor )
or
G.C.M. (Greatest common Multiple )
The H.C.F. of two or more than two numbers is greatest number that divides each one of them exactly. This is also Known as greatest cimmon divisor (G.C.D. ) or greatest common measure (G.C.M.) .
To Find H.C.F. of Given Numbers :
1) H.C.F. By Factorisation :
Express each number as the product of primes . Now, take the product of least powers of common factors to get the H.C.F. .
Example :
a) Find the H.C.F. of 96 , 528 and 792 .
Sol.
96 = 2X2X2X2X2X3
528 = 2X2X2X2X3X11
792 = 2X2X2X3X3X11
H.C.F. = 2X2X2X3
= 24
2) H.C.F. By Division :
Divide larger number by smaller one . Now , divide the divisor by the remainder . Repeat the process of the preceding divisor by the remainder last obtained , till the remainder 0 is obtained . The last divisor is the required H.C.F.
Example :
1) Find the H.C.F. of 370 and 592 .
Sol.
370 ) 592 ( 1
- 370
----------------
222 ) 370 ( 1
- 222
------------
148 ) 222 ( 1
- 148
------------
74 ) 148 ( 2
- 148
-----------
000
H.C.F. of given numbers = 74
2) Find the H.C.F. of 312 , 351 and 650 .
Sol.
312 ) 351 ( 1
- 312
-----------
39 ) 312 ( 8
- 312
-----------
000
39 ) 650 ( 16
- 39
--------
260
- 234
------
26 ) 39 ( 1
- 26
------
13 ) 39 ( 3
- 39
---------
00
H.C.F. of given numbers = 13
777
3) Reduce ---------- to lowest terms .
1147
Sol .
H.C.F. of 777 & 1147 is 37
On dividing Nr & Dr of the given fraction by 37 , we get :
777 21
= ------- = --------
1147 31
IV) L.C.M. ( Lowest common multiple )
The least number which is exactly divisible by each one of the given numbers , is called their L.C.M.
V) L.C.M. By Fatorization :
Resolve each one of the given numbers into prime factors . Then , the product of hightest powers of all the factors , gives the
L.C.M.
Example :
1) Find the L.C.M. of 96 ,108 & 280.
Sol.
96 = 2X2X2X2X2X3
108 = 2X2X3X3X3
280 = 2X2X2X5X7
L.C.M. = 2X2X2X2X2X3X3X3X5X7
= 30240
2) Find the L.C.M. of 12 , 15 , 18 , 27.
Sol.
2 | 12 - 15 - 18 - 27
--------|---------------------------------------------
3 | 6 - 15 - 9 - 27
---------|---------------------------------------------
3 | 2 - 5 - 3 - 9
---------|---------------------------------------------
| 2 - 5 - 1 - 3
so that
L.C.M. = 2X 3X 3X 2X 5X 3
= 540
VI) Two Important Rules :
1) Product of Two Numbers
= ( Their H.C.F. ) X ( Their L.C.M. )
2) H.C.F. of given numbers always divides their L.C.M.
Example :
Find the L.C.M. of 852 and 1491 .
Sol.
H.C.F. of 852 and 1491 = 213.
Product of Numbers
L.C.M. = -------------------------------------
H.C.F.
852 X 1491
= ----------------------------
213
= 5964
VII) H.C.F. & L.C.M. of fraction
1) H.C.F. of given fractions
H.C.F. of Numerators
= ------------------------------------------
L.C.M. of Denominators
2) L.C.M. of given fractions
L.C.M. of Numerators
= -----------------------------------------
H.C.F. of Denominators
Example:
Find the H.C.F. and L.C.M. of 8/9 ,32/81 ,and 10/27
Sol.
H.C.F. of 8, 32, 10 2
H.C.F. = ------------------------------- = -----
L.C.M. of 9, 81, 27 81
L.C.M. of 8, 32, 10 160
L.C.M. = ------------------------------- = ------------
H.C.F of 9, 81, 27 9
Exercise
1) The ratio of two numbers is 3 : 4 and their H.C.F. is 4 .Their L.C.M. is :
a) 12 b) 16 c) 24 d) 48
2) Three numbers are in the ratio 1: 2 : 3 and their H.C.F. is 12 The numbers are:
a) 4, 8, 12 b) 5, 10 ,15 c) 10, 20, 30
d) 12, 24, 36
3) The H.C.F. of 1.75 , 5.6 and 7 is :
a) 0.07 b) 0.7 c) 3.5 d) 0.35
4) The G.C.M. of 3556 and 3444 is :
a) 25 b) 28 c) 3 d) 26
5) H.C.F. of 204 , 1190 and 1445 is :
a) 16 b) 17 c) 18 d) 17.5
Answer 1) d 2) d 3) d 4) b
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