LCM and HCF
 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?

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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 cm^{2}
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 = 41Correct 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 cm^{2}
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
 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 :

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Required maximum capacity of container = HCF of 75L and 45L
Now, 75 = 5 × 5 × 3
45 = 5 × 3 × 3
∴ HCF = 15Correct 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
 What is the greatest number which will divide 110 and 128 leaving a remainder 2 in each case ?

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

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

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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 = 127Correct 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.