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Helium
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What is the smallest possible common multiple of three distinct Updated on: 29 Sep 2018, 23:51
Originally posted by Helium on 29 Sep 2018, 04:03.
Last edited by Helium on 29 Sep 2018, 23:51, edited 1 time in total.
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C
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Difficulty:

95% (hard)

Question Stats:

27% (01:37) correct 73% (02:01) wrong based on 89 sessions

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What is the smallest possible common multiple of three distinct integers, all larger than 26?

A) 1512
B) 729
C) 84
D) 2268
E) 108
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C
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What is the smallest possible common multiple of three distinct 29 Sep 2018, 04:25
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Solution



To find:
    • The smallest possible common multiple of three distinct integers, all larger than 26

Approach and Working:
The least number larger that 26 is 27.

To minimize the value of the common multiple, we should take other numbers also multiple of 27.
    • If we take 27, 54, and 81, their LCM is 162.
    • However, if we take 27, 54, and 108, their LCM is 108 only.

Hence, the correct answer is option C.

Answer: C

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What is the smallest possible common multiple of three distinct 29 Sep 2018, 05:53
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The smallest integer greater than 26 is 27.

The temptation is to go for LCM (26, 27, 28). On second thoughts, there are other better options.

If we can find 3 numbers such that they are 27a, 27b, 27c, the LCM will be 27 * LCM(a, b, c), where a, b, and c are distinct positive integers.
The lower the value of LCM(a, b, c), lower will be the value of the LCM of 27a, 27b, and 27c

Again the temptation is to take a, b, and c as 1, 2, and 3. The LCM of these 3 numbers is 6.
What if the 3 numbers are 1, 2, and 4. The LCM of these 3 numbers is 4.
We can infer that the least LCM possible for 3 distinct positive integers is 4. If one can remember this result, we can use it for posterity as well.

Therefore, the required LCM will be 27 * LCM(1, 2, 4) = 27 * 4 = 108
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What is the smallest possible common multiple of three distinct 29 Sep 2018, 23:20
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A rejoinder to my earlier discussion

For the given 5 choices, 108 is the least number.

However, the smallest possible common multiple of 3 distinct integers greater than 26 is 84 and NOT 108.

Reason
We started with 27 as the smallest integer greater than 26. Therefore, we went with 27a, 27b, and 27c being 27, 27*2 and 27*4. a, b, and c had to be integers. Else, 27, 27a, and 27b will not be integers.

What if we started with 28. Because 28 is an even number, our a, b, and c need not be integers.
We could opt for a = 1, b = 1.5 and c = 3. The LCM of 1, 1.5 and 3 is 3, which is lesser than the LCM of 1, 2, and 4
And because, 28 is even 28 * 1.5 will be an integer.
i.e., the 3 numbers could be 28, 42 and 84. The LCM will be 84.

If you are not convinced about a, b, and c being 1, 1.5 and 3, instead of considering it as 28a, 28b, 28c, think of the numbers as 14x, 14y, 14z where x, y, and z are 2, 3 and 6.
14*2 = 28, 14*3 = 42 and 14*6 = 84, all of which are greater than 26.

84 is the smallest possible common multiple of 3 distinct numbers greater than 26.
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What is the smallest possible common multiple of three distinct 29 Sep 2018, 23:50
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WizakoBaskar
A rejoinder to my earlier discussion

For the given 5 choices, 108 is the least number.

However, the smallest possible common multiple of 3 distinct integers greater than 26 is 84 and NOT 108.

Reason
We started with 27 as the smallest integer greater than 26. Therefore, we went with 27a, 27b, and 27c being 27, 27*2 and 27*4. a, b, and c had to be integers. Else, 27, 27a, and 27b will not be integers.

What if we started with 28. Because 28 is an even number, our a, b, and c need not be integers.
We could opt for a = 1, b = 1.5 and c = 3. The LCM of 1, 1.5 and 3 is 3, which is lesser than the LCM of 1, 2, and 4
And because, 28 is even 28 * 1.5 will be an integer.
i.e., the 3 numbers could be 28, 42 and 84. The LCM will be 84.

If you are not convinced about a, b, and c being 1, 1.5 and 3, instead of considering it as 28a, 28b, 28c, think of the numbers as 14x, 14y, 14z where x, y, and z are 2, 3 and 6.
14*2 = 28, 14*3 = 42 and 14*6 = 84, all of which are greater than 26.

84 is the smallest possible common multiple of 3 distinct numbers greater than 26.

Very insightful ... :thumbup:

Updated the options accordingly.
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What is the smallest possible common multiple of three distinct 03 Oct 2018, 06:52
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WizakoBaskar
A rejoinder to my earlier discussion

For the given 5 choices, 108 is the least number.

However, the smallest possible common multiple of 3 distinct integers greater than 26 is 84 and NOT 108.

Reason
We started with 27 as the smallest integer greater than 26. Therefore, we went with 27a, 27b, and 27c being 27, 27*2 and 27*4. a, b, and c had to be integers. Else, 27, 27a, and 27b will not be integers.

What if we started with 28. Because 28 is an even number, our a, b, and c need not be integers.
We could opt for a = 1, b = 1.5 and c = 3. The LCM of 1, 1.5 and 3 is 3, which is lesser than the LCM of 1, 2, and 4
And because, 28 is even 28 * 1.5 will be an integer.
i.e., the 3 numbers could be 28, 42 and 84. The LCM will be 84.

If you are not convinced about a, b, and c being 1, 1.5 and 3, instead of considering it as 28a, 28b, 28c, think of the numbers as 14x, 14y, 14z where x, y, and z are 2, 3 and 6.
14*2 = 28, 14*3 = 42 and 14*6 = 84, all of which are greater than 26.

84 is the smallest possible common multiple of 3 distinct numbers greater than 26.

What if we consider least three integer after 26, i.e. 27,28 and 29, and break it down to its prime factors. We will find that the common multiple must have 2,3 and 7 among its multiples. Which renders option B and E useless. Now between remaining options we can select the smallest but it still doesn't answer about 29, which is a prime number. Could you please elaborate if my reasoning is correct ?
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What is the smallest possible common multiple of three distinct 03 Oct 2018, 09:14
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suelahmed
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A rejoinder to my earlier discussion

For the given 5 choices, 108 is the least number.

However, the smallest possible common multiple of 3 distinct integers greater than 26 is 84 and NOT 108.

Reason
We started with 27 as the smallest integer greater than 26. Therefore, we went with 27a, 27b, and 27c being 27, 27*2 and 27*4. a, b, and c had to be integers. Else, 27, 27a, and 27b will not be integers.

What if we started with 28. Because 28 is an even number, our a, b, and c need not be integers.
We could opt for a = 1, b = 1.5 and c = 3. The LCM of 1, 1.5 and 3 is 3, which is lesser than the LCM of 1, 2, and 4
And because, 28 is even 28 * 1.5 will be an integer.
i.e., the 3 numbers could be 28, 42 and 84. The LCM will be 84.

If you are not convinced about a, b, and c being 1, 1.5 and 3, instead of considering it as 28a, 28b, 28c, think of the numbers as 14x, 14y, 14z where x, y, and z are 2, 3 and 6.
14*2 = 28, 14*3 = 42 and 14*6 = 84, all of which are greater than 26.

84 is the smallest possible common multiple of 3 distinct numbers greater than 26.

What if we consider least three integer after 26, i.e. 27,28 and 29, and break it down to its prime factors. We will find that the common multiple must have 2,3 and 7 among its multiples. Which renders option B and E useless. Now between remaining options we can select the smallest but it still doesn't answer about 29, which is a prime number. Could you please elaborate if my reasoning is correct ?


Mate please go thorough the discussion above..it already has answer for your query...

Re-read the question prompt - i.e. smallest possible common multiple of three distinct integers, all larger than 26.

So we are looking for smallest LCM possible with 3 distinct numbers greater than 26.

Numbers need not be 27, 28 and 29.

As discussed in this thread above..

Numbers we are looking for are 28, 42 and 84. ( These are 3 distinct numbers and also 84 is smallest possible common multiple.)

Does this help?
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What is the smallest possible common multiple of three distinct 03 Oct 2018, 09:50
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WizakoBaskar
A rejoinder to my earlier discussion

For the given 5 choices, 108 is the least number.

However, the smallest possible common multiple of 3 distinct integers greater than 26 is 84 and NOT 108.

Reason
We started with 27 as the smallest integer greater than 26. Therefore, we went with 27a, 27b, and 27c being 27, 27*2 and 27*4. a, b, and c had to be integers. Else, 27, 27a, and 27b will not be integers.

What if we started with 28. Because 28 is an even number, our a, b, and c need not be integers.
We could opt for a = 1, b = 1.5 and c = 3. The LCM of 1, 1.5 and 3 is 3, which is lesser than the LCM of 1, 2, and 4
And because, 28 is even 28 * 1.5 will be an integer.
i.e., the 3 numbers could be 28, 42 and 84. The LCM will be 84.

If you are not convinced about a, b, and c being 1, 1.5 and 3, instead of considering it as 28a, 28b, 28c, think of the numbers as 14x, 14y, 14z where x, y, and z are 2, 3 and 6.
14*2 = 28, 14*3 = 42 and 14*6 = 84, all of which are greater than 26.

84 is the smallest possible common multiple of 3 distinct numbers greater than 26.

What if we consider least three integer after 26, i.e. 27,28 and 29, and break it down to its prime factors. We will find that the common multiple must have 2,3 and 7 among its multiples. Which renders option B and E useless. Now between remaining options we can select the smallest but it still doesn't answer about 29, which is a prime number. Could you please elaborate if my reasoning is correct ?


Mate please go thorough the discussion above..it already has answer for your query...

Re-read the question prompt - i.e. smallest possible common multiple of three distinct integers, all larger than 26.

So we are looking for smallest LCM possible with 3 distinct numbers greater than 26.

Numbers need not be 27, 28 and 29.

As discussed in this thread above..

Numbers we are looking for are 28, 42 and 84. ( These are 3 distinct numbers and also 84 is smallest possible common multiple.)

Does this help?


ok... thanks for xplaination.. :)
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What is the smallest possible common multiple of three distinct 03 Feb 2020, 09:05
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