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Re: What is the smallest integer that is divisible by 4, 12, and 15 at the [#permalink]
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PGTLrowanhand wrote:
You can easily test for Least Common Multiple (the smallest integer divisible) with a simple process.

First, imagine multiplying all the numbers together: you'd get a big number (OK, it's 720). The reason that there exist smaller numbers than this one that actually do divide is this: there exist redundant factors among 4, 12, and 15. For example, 4 and 12 both divide by 4. 12 and 15 both divide by 3, etc. The numbers are redundant because we don't need to specify that their multiple is divisible, for example, by 4 two times--in fact, that would be saying that it is divisible by 8. That is precisely the outcome we are trying to avoid.

What we want to do is to list out all the factors of each number, individually, and then eliminate all of the redundant factors. If we multiply then, we'll see the smallest number divisible by each of 4, 12, and 15. This would work for any number of numbers, of course.

The process I use is as follows:

1) Prime Factorize:

4: 2^2

12: 2^2 * 3

15: 3 *5

2) Eliminate redundancy: scratch out any number that appears more than once, keeping the highest power of each base.

In this case we get...

4: ___ (nothing)

12: 2^2 * ___ (3 eliminated)

15: 3*5

3) Multiply: the answer is then the remaining numbers multiplied...

2^2 * 3 * 5 = 4 * 15 = 60

Answer B.

And I mean look--you can always just test the numbers. However if you are testing ten numbers instead of three, a process like this will make your life substantially easier. It's worth learning because when you're good at it this is even faster than testing.


Great reply
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Re: What is the smallest integer that is divisible by 4, 12, and 15 at the [#permalink]
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GreenlightTestPrep

30 is not divisible by 4. 30 is divisible by the integers: 1, 2, 3, 5, 6, 10, 15, and 30.
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Re: What is the smallest integer that is divisible by 4, 12, and 15 at the [#permalink]
superpower101 wrote:
GreenlightTestPrep

30 is not divisible by 4. 30 is divisible by the integers: 1, 2, 3, 5, 6, 10, 15, and 30.


My bad!
I have edited my solution accordingly.
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Re: What is the smallest integer that is divisible by 4, 12, and 15 at the [#permalink]
1
Smallest integer that is divisible by 4, 12, and 15 will be nothing but the least Common Multiple (LCM) of these numbers.

So, we need to find LCM(4, 12, 15).

Solved using two Methods

Method 1: Long Division Method

4 | 4,12,15
-------------------
3 | 1,3,15
-------------------
5 | 1,1,5
-------------------
| 1,1,1

=> LCM(4, 12, 15) = 4*3*5 = 60

So, Answer will be B.
Hope it helps!

Method 2: Prime Factorization Method

Write each number of power of prime number

LCM(4, 12, 15) = LCM(22,2231,3151)

To find LCM, take highest power of each prime number
=> LCM(4, 12, 15) = 223151= 4*3*5 = 60

So, Answer will be B.
Hope it helps!

To learn more about LCM and GCD watch the following videos



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Re: What is the smallest integer that is divisible by 4, 12, and 15 at the [#permalink]
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