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31. The HCF and LCM of two numbers are 8 and 48 respectively. If one of the number is 24, then the other number is = ?

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Solution:
HCF = 8
LCM = 48
One number = 24
Let other number be = y
∴ 24y = 48 × 8
⇔ y = 16
32. The least number which when divided by 4, 6, 8, 12 and 16 leaves remainder of 2 in each case is = ?

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Solution:
LCM of (4, 6, 8, 12, 16)
⇒ 16 × 3 = 48
∴ The number when divided by (4, 6, 8, 12, 16) leaves remainder 2 is
= 48 + 2
= 50
33. The LCM of two numbers is 495 and their HCF is 5. If the sum of the numbers is 100, then their difference is = ?

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Solution:
LCM = 495
HCF = 5       ( given )
∴ Let numbers are = 5x and 5y
∴ LCM = 5xy
⇒ 5xy = 495
⇒ xy = 99
∴ Possible co-prime factors are
1,       99
9,       11
∴ Possible numbers are 5x, 5y
45,       55
5,       495
Now given that sum of numbers = 100
So, required numbers are = (45, 55)
∴ Difference of numbers = 55 - 45 = 10
34. Which of the following has most number of divisors?

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Solution:
99 = 1 × 3 × 3 × 11
101 = 1 × 101
176 = 1 × 2 × 2 × 2 × 2 × 11
182 = 1 × 2 × 7 × 13
So, divisor of 99 are 1, 3, 9, 11, 33 and 99
Divisor of 101 are 1 and 101
Divisor of 176 are 1, 2, 4, 8, 11, 16, 22, 44, 88 and 176
Divisor of 182 are 1, 2, 7, 13, 14, 26, 91 and 182
Hence, 176 has the most number of divisors.
35. The HCF of two numbers is 8. Which one of the following can never be their LCM ?

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Solution:
HCF = 8
Now LCM should have a factors 8
So check also the option we have only 60 which does not have a factor 8. So it will never be the LCM.
36. The LCM and the HCF of the numbers 28 and 42 are in the ratio ?

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Solution:
Numbers x = 28, y = 42
HCF (28 , 42)
Difference = 42 - 28 = 14
For HCF of any numbers take their difference. HCF will be either the factor of that difference or the difference itself.
Now, LCM of 28, 42
∴ 14 × 2 × 3 = 84
⇒ LCM : HCF
⇒ 84 : 14
⇒ 6 : 1
37. The simplest reduction to the lowest terms of 116690151427863887   is = ?

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Solution:
116690151)427863887(¯335007045377793434)116690151(¯17779343438896717)77793434(¯277793434×So, H.C.F of 116690151 and 427863887 = 38896717∴116690151427863887=116690151÷38896717427863887÷38896717=311
38. The LCM of two numbers is 1820 and their HCF is 26. If one number is 130 then the other number is ?

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Solution:
LCM = 1820HCF = 261st number = 130LCM×HCF  = Product of numbers⇒Let the other number is x 130×x=1820×26x=1820×26130=364
39. The HCF of two number 12906 and 14818 is 478. Their LCM is ?

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Solution:
HCF = 478
Number are = 12906 and 14818
LCM × HCF = 12906 × 14818
LCM × 478 = 12906 × 14818
LCM = 400086
40. Find the greatest number of five digits which when divided by 3, 5, 8, 12 leaves 2 as remainder ?

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Solution:
LCM of (3, 5, 8, 12)
⇒ 3 × 5 × 4 × 2 = 120
⇒ Now greatest five digits number is 99999
On dividing 99999 by = 120 (LCM)
We get remainder = 99999120,  remainder = 39
⇒ By subtracting remainder from 99999 we get the greatest five digits number which is completely divisible by given numbers (3, 5, 8, 12)
∴ 99999 - 39 = 99960
⇒ Now, we required greatest five digit number which when divided by (3, 5, 8, 12) leaves remainder 2 in each case.
⇒ add 2 in the 99960
⇒ 99960 + 2
⇒ 99962

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