LCM and HCF
 Let N be the greatest number that will divide 1305, 4665 and 6905 leaving the same remainder in each case. Then, sum of the digits in N is :

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We can say 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).
The greatest number N = HCF of (1305 – t ), (4665 – t ) and (6905 – t), where t is the remainder
= HCF of (4665 – 1305), (6905– 4665) and (6905 – 1305)
= HCF of 3360, 2240 and 5600Correct Option: A
We can say 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).
The greatest number N = HCF of (1305 – t ), (4665 – t ) and (6905 – t), where t is the remainder
= HCF of (4665 – 1305), (6905– 4665) and (6905 – 1305)
= HCF of 3360, 2240 and 5600
∴ N = 1120
Sum of digits = 1 + 1 + 2 + 0 = 4
 What is the greatest number that will divide 307 and 330 leaving remainders 3 and 7 respectively ?

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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).
The number will be HCF of 307 – 3 = 304 and 330 – 7 = 323.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).
The number will be HCF of 307 – 3 = 304 and 330 – 7 = 323.
∴ Required number = 19
 Which greatest number will divide 3026 and 5053 leaving remainders 11 and 13 respectively?

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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).
3026 –11 = 3015 and 5053 –13 = 5040
Required number = HCF of 3015 and 5040Correct Option: C
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).
3026 –11 = 3015 and 5053 –13 = 5040
Required number = HCF of 3015 and 5040
∴ Required number = HCF of 3015 and 5040 = 45
 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.
 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