SOLVED EXAMPLES

Ex. 1. Find the H.C.F. of 2³  X 3²   X 5 X 7⁴, 2²  X 3⁵ X 5²  X 7³,2³ X 5³  X 7²

Sol.    

The prime numbers common to given numbers are 2,5 and 7.

H.C.F. = 2² x 5 x7² = 980.

Ex. 2. Find the H.C.F. of 108, 288 and 360.

Sol.    

108 = 2² x 3³, 288 = 2⁵ x 3² and 360 = 2³ x 5 x 3².

H.C.F. = 2² x 3² = 36.

Ex. 3. Reduce 391     to lowest terms.

                        667

Sol.    

H.C.F. of 391 and 667 is 23.

On dividing the numerator and denominator by 23, we get :

391 = 391 ¸ 23 = 17

667    667¸ 23    29

Ex.4.Find the L.C.M. of 2² x 3³ x 5 x 7^{2 ,} 2³ x 3² x 5² x 7⁴  , 2 x 3 x 5^{3 x 7 x 11.}  

Sol. 

L.C.M. = Product of highest powers of 2, 3, 5, 7 and 11 = 2³ x 3³ x 5³ x 7⁴ x 11

Ex.5. Find the L.C.M. of 72, 108 and 2100.

Sol. 

72 = 2³ x 3², 108 = 3³ x 2², 2100 = 2² x 5² x 3 x 7.

 L.C.M. = 2³ x 3³ x 5² x 7 = 37800.

Ex.6.Find the L.C.M. of 16, 24, 36 and 54.

Sol.

2	16	-   24	-   36	-   54
2	8	-   12	-   18	-   27
2	4	-     6	-     9	-   27
3	2	-     3	-     9	-   27
3	2	-     1	-     3	-     9
	2	-     1	-     1	-     3

\ L.C.M. = 2 x 2 x 2 x 3 x 3 x 2 x 3 = 432.

Ex. 7. Find the H.C.F. and L.C.M. of 2  , 8  , 16 and 10.

                                                               3    9    81        27   

Sol.    H.C.F. of given fractions = H.C.F. of 2,8,16,10   =     2_   

                                                       L.C.M. of 3,9,81,27        81

          L.C.M of given fractions = L.C.M. of 2,8,16,10   =    80_   

                                                      H.C.F. of 3,9,81,27           3

Ex. 8. Find the H.C.F. and L.C.M. of 0.63, 1.05 and 2.1.

Sol.     

Making the same number of decimal places, the given numbers are 0.63, 1.05 and 2.10.

Without decimal places, these numbers are 63, 105 and 210.

Now, H.C.F. of 63, 105 and 210 is 21.

H.C.F. of 0.63, 1.05 and 2.1 is 0.21.

L.C.M. of 63, 105 and 210 is 630.

L.C.M. of 0.63, 1.05 and 2.1 is 6.30.

Ex. 9. Two numbers are in the ratio of 15:11. If their H.C.F. is 13, find the numbers.

Sol.    

Let the required numbers be 15.x and llx.

Then, their H.C.F. is x. So, x = 13.

The numbers are (15 x 13 and 11 x 13) i.e., 195 and 143.

Ex. 10. TheH.C.F. of two numbers is 11 and their L.C.M. is 693. If one of the numbers is 77,find the other.

Sol.    

Other number = 11 X 693   = 99

                             77

Ex.11.Find the least number exactly divisible by 12,15,20,27.

Sol.

           

3	12	-   15	-   20	-   27
4	4	-   5	-   20	-   9
5	1	-   5	-     5	-   9
	1	-    1	-     1	-   9

Ex.12.Find the least number which when divided by 6,7,8,9, and 12 leave the same remainder 1 each case

Sol. 

Required number = (L.C.M OF 6,7,8,9,12) + 1

3	6	-   7	-    8	-   9    -   12
4	2	-   7	-   8	-   3   -   4
5	1	-   7	-     4	-   3   -   2
	1	-    7	-     2	-   3   -   1

\L.C.M = 3 X 2 X 2 X 7 X 2 X 3 = 504.

Hence required number = (504 +1) = 505.

The End