## LCM and HCF

#### LCM and HCF

1. What is the least number of square tiles required to pave the floor of a room 15 m 17 cm long and 9 m 2 cm broad?

1. Length of the floor = 15 m 17 cm = 1517 cm
Breadth of the floor = 9m 2 cm = 902 cm.
Area of the floor = 1517 × 902 cm2
The number of square tiles will be least, when the size of each tile is maximum.
∴ Size of each tile = HCF of 1517 and 902 = 41

##### Correct Option: D

Length of the floor = 15 m 17 cm = 1517 cm
Breadth of the floor = 9m 2 cm = 902 cm.
Area of the floor = 1517 × 902 cm2
The number of square tiles will be least, when the size of each tile is maximum.
∴ Size of each tile = HCF of 1517 and 902 = 41

 ∴ Required number of tiles = 1517 × 902 = 814 41 × 41

1. A milkman has 75 litres milk in one can and 45 litres in another. The maximum capacity of container which can measure milk of either container exact number of times is :

1. Required maximum capacity of container = HCF of 75L and 45L
Now, 75 = 5 × 5 × 3
45 = 5 × 3 × 3
∴ HCF = 15

##### Correct Option: C

Required maximum capacity of container = HCF of 75L and 45L
Now, 75 = 5 × 5 × 3
45 = 5 × 3 × 3
∴ HCF = 15
Required maximum capacity of container = 15 litres

1. What is the greatest number which will divide 110 and 128 leaving a remainder 2 in each case ?

1. As we know that the largest number which when divide the numbers a, b and c give remainders as p, q, r respectively is given by H.C.F. of (a – p), (b – q) and (c – r).
Required number = HCF of (110 – 2) and (128 – 2)

##### Correct Option: B

As we know that the largest number which when divide the numbers a, b and c give remainders as p, q, r respectively is given by H.C.F. of (a – p), (b – q) and (c – r).
Required number = HCF of (110 – 2) and (128 – 2)
Required number = HCF of 108 and 126 = 18

1. The largest number, which divides 25, 73 and 97 to leave the same remainder in each case, is

1. We know that the largest number which when divide the numbers a, b and c give remainders as p, q, r respectively is given by H.C.F. of (a – p), (b – q) and (c – r).
Let t be the remainder.
Then, (25 – t ), (73 – t ), and (97 – t ) Will be exactly divisible by the required number.
∴ Required number = HCF of (73 – t ) – (25 – t ), (97 – t ) – (73 – t ) and (97 – t ) – (25 – t )

##### Correct Option: A

We know that the largest number which when divide the numbers a, b and c give remainders as p, q, r respectively is given by H.C.F. of (a – p), (b – q) and (c – r).
Let t be the remainder.
Then, (25 – t ), (73 – t ), and (97 – t ) Will be exactly divisible by the required number.
∴ Required number = HCF of (73 – t ) – (25 – t ), (97 – t ) – (73 – t ) and (97 – t ) – (25 – t )
Required number = HCF of (73 – 25), (97 – 73), and (97 – 25) = HCF of 48, 24 and 72 = 24

1. The greatest number, by which 1657 and 2037 are divided to give remainders 6 and 5 respectively, is

1. As we know that the largest number which when divide the numbers a, b and c give remainders as p, q, r respectively is given by H.C.F. of (a – p), (b – q) and (c – r).
We have to find HCF of (1657 – 6 = 1651) and
(2037 – 5 = 2032)
1651 = 13 × 127
2032 = 16 × 127
∴ HCF = 127

##### Correct Option: A

As we know that the largest number which when divide the numbers a, b and c give remainders as p, q, r respectively is given by H.C.F. of (a – p), (b – q) and (c – r).
We have to find HCF of (1657 – 6 = 1651) and
(2037 – 5 = 2032)
1651 = 13 × 127
2032 = 16 × 127
∴ HCF = 127
So, required number will be 127.