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41. The H.C.F of 24 × 32 × 53 × 7, 23 × 33 × 52 × 72 and 3 × 5 × 7 × 11 is = ?

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Solution:
H.C.F. = Product of lowest powers of common factors
           = 3 × 5 × 7
           = 105
42. The H.C.F of 4 × 27 × 3125, 8 × 9 × 25 × 7 and 16 × 81 × 5 × 11 × 49 is = ?

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Solution:
4 × 27 × 3125 = 22 × 33 × 55
8 × 9 × 25 × 7 = 23 × 32 × 52 × 7
16 × 81 × 5 × 11 × 49 = 24 × 34 × 5 × 11 × 72
∴ H.C.F = 22 × 32 × 5 = 180
43. Even numbers are formed by talking at least two at a time from the numbers 0, 4, 8, 9. Their H.C.F. is = ?

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Solution:
Since all the numbers formed are even, 2 is a common factor.
Also, H.C.F. of two of the numbers i.e., 48 and 490, is 2.
So, the H.C.F. of all the numbers formed is 2.
44. Find the largest number of four digits such that on dividing by 15, 18, 21 and 24 the remainders are 11, 14, 17 and 20 respectively.

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Solution:
LCM of (15, 18, 21, 24)
⇒ 5 × 3 × 6 × 7 × 4 = 2520
⇒ In such type of questions,we take the difference between given number and remainder of that number.
Number Remainder
⇒ (15 - 11) = 4
⇒ (18 - 14) = 4
⇒ (21 - 17) = 4
⇒ (24 - 20) = 4
      It will be same always
Now, Largest 4 digit number is 9999
⇒ On dividing 9999 by LCM (2520) we get remainder
⇒ 2439
Subtract remainder from 9999 we get largest 4 digit number, which is divisible by given number
= 9999 - 2439 = 7560 (9999 but required number gives difference on dividing)
So, our required number = 7560 - 4(difference) = 7556
45. The greatest number which when divides 989 and 1327 leave remainder 5 and 7 respectively = ?

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Solution:
989 - 5 = 984
1327 - 7 = 1320
Subtract the remainder from the number.
HCF = (984, 1320) = 24
for greatest number take HCF of the numbers.
46. A milkman has 75 litres milk in one cane and 45 litres in another. The maximum capacity of container which can measure milk of either container exact number ?

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Solution:
75 litres, 45 litres
For maximum capacity take HCF (75, 45) = 15
47. Which of the following is a pair of co-primes ?

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Solution:
H.C.F. of 18 and 25 is 1
So, they are co-primes
48. The L.C.M. of 22, 54, 108, 135 and 198 is = ?

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Solution:
 2   22   -   54   -   108   -   135   -   198 
 3   11   -   27   -   54   -   135   -   99 
 3   11   -   9   -   18   -   45   -   33 
 3   11   -   3   -   6   -   15   -   11 
 11   11   -   1   -   2   -   5   -   11 
   1   -   1   -   2   -   5   -   1 

L.C.M. = 2 × 3 × 3 × 3 × 11 × 2 × 5
            = 5940
49. Two numbers are in the ratio 3 : 4, if their H.C.F. is 4, then their L.C.M. = ?

Discuss
Solution:
Let number be = x, y
x : y = 3 : 4 (given)
H.C.F. = 4
∴ Number are = x = 4 × 3 = 12
y = 4 × 4 = 16
L.C.M. of (12, 16)
= 4 × 3 × 4
= 48
50. L.C.M. of two numbers is 2079 and their H.C.F. is 27. If one of the number is 189, the other number is = ?

Discuss
Solution:
Given,
H.C.F. = 27
L.C.M. = 2079
one number = 189
Let another number be y
⇒ Product of numbers = L.C.M. × H.C.F.
∴ 189 x y = 27 × 2079
y = 297

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