H.C.F. and L.C.M.

1. Two numbers are in the ratio 2 : 3. If their L.C.M. is 48. what is sum of the numbers?
A. 28	B. 40
C. 64	D. 42

Answer : Option B

Explanation :

Let the numbers be 2x and 3x

LCM of 2x and 3x = 6x (∵ LCM of 2 and 3 is 6. Hence LCM

of 2x and 3x is 6x)

Given that LCM of 2x and 3x is 48

=> 6x = 48

=> x = ⁴⁸⁄₆ = 8

Sum of the numbers = 2x + 3x = 5x = 5 × 8 = 40

2. What is the greatest number of four digits which is divisible by 15, 25, 40 and 75 ?
A. 9800	B. 9600
C. 9400	D. 9200

Answer : Option B

Explanation :

Greatest number of four digits = 9999

LCM of 15, 25, 40 and 75 = 600

9999 ÷ 600 = 16, remainder = 399

Hence, greatest number of four digits which is divisible by 15, 25, 40 and 75

= 9999 - 399 = 9600

3. Three numbers are in the ratio of 2 : 3 : 4 and their L.C.M. is 240. Their H.C.F. is:
A. 40	B. 30
C. 20	D. 10

Answer : Option C

Explanation :

Let the numbers be 2x, 3x and 4x

LCM of 2x, 3x and 4x = 12x

=> 12x = 240

=> x = ²⁴⁰⁄₁₂ = 20

H.C.F of 2x, 3x and 4x = x = 20

4. What is the lowest common multiple of 12, 36 and 20?
A. 160	B. 220
C. 120	D. 180

Answer : Option D

Explanation :

  2 

12, 36, 20
2 

6,  18, 10
3 

3,  9,   5
1,  3,  5 

LCM = 2 × 2 × 3 × 1 × 3 × 5 = 180

5. What is the least number which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder?
A. 1108	B. 1683
C. 2007	D. 3363

Answer : Option B

Explanation :

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Solution 1
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LCM of 5, 6, 7 and 8 = 840

Hence the number can be written in the form (840k + 3) which is divisible by 9

If k = 1, number = (840 × 1) + 3 = 843 which is not divisible by 9

If k = 2, number = (840 × 2) + 3 = 1683 which is divisible by 9

Hence 1683 is the least number which when divided by 5, 6, 7 and 8 leaves a remainder 3,

but when divided by 9 leaves no remainder

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Solution 2 - Hit and Trial Method
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Just see which of the given choices satisfy the given condtions

Take 3363. This is not even divisible by 9. Hence this is not the answer

Take 1108. This is not even divisible by 9. Hence this is not the answer

Take 2007. This is divisible by 9.

2007 ÷ 5 = 401, remainder = 2 . Hence this is not the answer

Take 1683. This is divisible by 9.

1683 ÷ 5 = 336, remainder = 3

1683 ÷ 6 = 280, remainder = 3

1683 ÷ 7 = 240, remainder = 3

1683 ÷ 8 = 210, remainder = 3

Hence 1683 is the answer