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The L.C.M. of three different numbers is 120. Which of the following cannot be their H.C.F.?
- a)8
- b)12
- c)24
- d)35
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
The L.C.M. of three different numbers is 120. Which of the following c...
We know that:
LCM is the least common multiple of the given numbers whereas HCF is the highest common factor of those numbers.
Then, LCM is the multiplication of one common factor of the numbers and the other different factors of the numbers.
Write the LCM = 120 into factored form, that is
120 = 2 × 2 × 2 × 3 × 5
= 4(2 × 3 × 5)
⇒ 4 is the common factor of the numbers.
So, the HCF of three numbers is a multiple of 4.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
Therefore, 35 is not the multiple of 4, then 35 cannot be their HCF.
LCM is the least common multiple of the given numbers whereas HCF is the highest common factor of those numbers.
Then, LCM is the multiplication of one common factor of the numbers and the other different factors of the numbers.
Write the LCM = 120 into factored form, that is
120 = 2 × 2 × 2 × 3 × 5
= 4(2 × 3 × 5)
⇒ 4 is the common factor of the numbers.
So, the HCF of three numbers is a multiple of 4.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
Therefore, 35 is not the multiple of 4, then 35 cannot be their HCF.
Most Upvoted Answer
The L.C.M. of three different numbers is 120. Which of the following c...
Understanding L.C.M. and H.C.F.
The Least Common Multiple (L.C.M.) is the smallest multiple that two or more numbers share, while the Highest Common Factor (H.C.F.) is the largest number that divides two or more numbers without leaving a remainder.
Given Information
- The L.C.M. of three different numbers is 120.
- We need to determine which of the given options cannot be their H.C.F.
Calculating Factors
1. Factors of 120:
- 120 can be factored into prime numbers:
- 120 = 2^3 × 3^1 × 5^1.
2. Possible H.C.F. Values:
- For the H.C.F. to be a valid factor, it must also divide the L.C.M. of 120.
Evaluating Options
- Option A: H.C.F. = 8
- 8 (2^3) divides 120. Valid.
- Option B: H.C.F. = 12
- 12 (2^2 × 3^1) divides 120. Valid.
- Option C: H.C.F. = 24
- 24 (2^3 × 3^1) divides 120. Valid.
- Option D: H.C.F. = 35
- 35 (5^1 × 7^1) does not divide 120, as 120 does not contain the factor 7.
Conclusion
Since the H.C.F. must be a divisor of the L.C.M., and 35 does not divide 120, it cannot be the H.C.F. of any three numbers whose L.C.M. is 120. Therefore, the correct answer is:
Option D: 35
The Least Common Multiple (L.C.M.) is the smallest multiple that two or more numbers share, while the Highest Common Factor (H.C.F.) is the largest number that divides two or more numbers without leaving a remainder.
Given Information
- The L.C.M. of three different numbers is 120.
- We need to determine which of the given options cannot be their H.C.F.
Calculating Factors
1. Factors of 120:
- 120 can be factored into prime numbers:
- 120 = 2^3 × 3^1 × 5^1.
2. Possible H.C.F. Values:
- For the H.C.F. to be a valid factor, it must also divide the L.C.M. of 120.
Evaluating Options
- Option A: H.C.F. = 8
- 8 (2^3) divides 120. Valid.
- Option B: H.C.F. = 12
- 12 (2^2 × 3^1) divides 120. Valid.
- Option C: H.C.F. = 24
- 24 (2^3 × 3^1) divides 120. Valid.
- Option D: H.C.F. = 35
- 35 (5^1 × 7^1) does not divide 120, as 120 does not contain the factor 7.
Conclusion
Since the H.C.F. must be a divisor of the L.C.M., and 35 does not divide 120, it cannot be the H.C.F. of any three numbers whose L.C.M. is 120. Therefore, the correct answer is:
Option D: 35
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