Integers: Finding the LCM - Least Common Multiple
Which of the following is the smallest number that is divisible by all the multiples of 2 between 1 and 9?
__Alternative Method__:
Since the answer choices hold quite small numbers, __Reverse PI__ will
also work well. Start with answer choice A since the question requires
the _smallest_ number that is divisible by 2, 4, 6, and 8:
For answer choice A, the number 16 is not divisible by 6. __POE__ and move on.
For answer choice B, 18 is not divisible by 8. __POE__ and move on.
Answer choice C is the smallest answer choice that is divisible by all 4 integers. Therefore, this is the correct answer.
Incorrect.
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Yes, 48 is divisible by all the multiples of 2 between 1 and 9. However, is it the smallest number which meets these criteria?
Incorrect.
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Eliminate this answer choice because it is not divisible by 8.
Incorrect.
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Eliminate this answer choice because it is not divisible by 6.
Incorrect.
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Eliminate this answer choice because it is not divisible by 6.
Correct.
List all the multiples of 2 between 1 and 9. They are 2, 4, 6, and 8.
In order to find the smallest number that is divisible by 2, 4, 6, and 8 – that is, the least common multiple (LCM) of 2, 4, 6, and 8 – you need to find the prime factors of these numbers:
>$$2 = 2$$
>$$4 = 2 \times 2$$
>$$6 = 2 \times 3$$
>$$8 = 2 \times 2 \times 2$$
Then construct a list of the minimum requirements for each factor.
Count the maximum number of times each prime factor occurs for any of
the four numbers. Keep the prime factor for the highest count and
ignore the rest. Now, multiply the remaining factors to calculate the
LCM.
The 2's highest count is in 8 ($$2 \times 2 \times 2$$), so cancel out 2 from 2, 4, and 6 since they do not have the highest count.
Now multiply the remaining factors to find the smallest multiple divisible by 2, 4, 6 and 8:
>$$\text{LCM} = 3 \times (2 \times2 \times 2) = 3 \times8 = 24$$.
Hence, this is the correct answer.
16
18
24
48
56
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