MHB Throwing in Towel: Finding an LCD for 186 and 300

[Duckfan]

I have been wrecking my brain to find an LCD for both 186 and 300. 30 works for 300 but not for 186-obviously. So what do I do in this situation? I find it baffling someone wrote a problem where only the denominator (300) has an LCD that fits. But I must be missing something other than a few brain cells. A few of the possibles/combinations I found were:

30x10 = 300
30x6 = 180

4x4x4=64x2=128
32x3=96x2=192
28x6=168
36x3=108
12x16=192

This has to be a trick problem? Yes?

 

[Prove It]

I have been wrecking my brain to find an LCD for both 186 and 300. 30 works for 300 but not for 186-obviously. So what do I do in this situation? I find it baffling someone wrote a problem where only the denominator (300) has an LCD that fits. But I must be missing something other than a few brain cells. A few of the possibles/combinations I found were:

30x10 = 300
30x6 = 180

4x4x4=64x2=128
32x3=96x2=192
28x6=168
36x3=108
12x16=192

This has to be a trick problem? Yes?

Write down ALL the factors of 300 and 186, and then pick the largest factor that appears in both.

 

[MarkFL]

I would use the prime factorization method to get the GCD.

$$300=(30)(10)=(2)(3)(5)(2)(5)=2^2\cdot3\cdot5^2$$

$$186=(2)(93)=2\cdot3\cdot31$$

Now we can see that the largest divisor of both is what?

 

[Duckfan]

I would use the prime factorization method to get the GCD.

$$300=(30)(10)=(2)(3)(5)(2)(5)=2^2\cdot3\cdot5^2$$

$$186=(2)(93)=2\cdot3\cdot31$$

Now we can see that the largest divisor of both is what?

Well, it looks like 2 is prime factor for this. I didn't expect I had to start that low. because as you can see I was busy playing with other combinations similar to what you used in other example. I was trying to work it out with what I was sure method to find LCD, but getting frustrated because I wasn't finding it. Just seemed an odd combination to work with. 300, I can deal with that. 186. A bit of a curve ball. Thank you MarkFL.

 

[MarkFL]

Do you see that $2\cdot3=6$ is the largest divisor in both numbers? :)

 

[Duckfan]

Do you see that $2\cdot3=6$ is the largest divisor in both numbers? :)

No. I'm sorry I didn't. I just saw 2. I thought I'm supposed to look for single "digit" or number that is common to both N & D.

Please don't shoot me for not seeing it.

And even if I did see it, I still probably would not have caught on to use it further since I've been programmed to look for a "single" number that fits into both N & D.

 

[MarkFL]

No. I'm sorry I didn't. I just saw 2. I thought I'm supposed to look for single "digit" or number that is common to both N & D.

Please don't shoot me for not seeing it.

What we want to do is first look at the prime factors (ignore the exponents for now) each number has. We see that the two numbers have the following prime factors:

300: 2, 3, 5

186: 2, 3, 31

So, we see that 2 and 3 are the only prime factors common to both. Now the exponents come into play (and recall a number without an exponent has an implied 1 as an exponent, i.e. $n=n^1$)...we take the smaller of the exponents on each prime factor (in the case of a "tie" we take the common exponent)...in this case we see for both factors, the smaller exponent is 1, and so now we may state:

$$\gcd(300,186)=2\cdot3=6$$

 

[Duckfan]

I just want you to know that I honestly took a lot of time trying to find the factor that goes into both 186, 300. I really did sat here and pulled my hair out wondering what I was missing. As I said, I honestly did not see 2 x 3. I'm trying to figure this on my own since the exam monitors would likely not be happy if I brought someone into help with question. I appreciate the breakdown. I really do. (Whew) Moving away from fractions now and getting into "real number" issues/problems.

And BTW, if you have Skype, I have it too.

 

[MarkFL]

I just want you to know that I honestly took a lot of time trying to find the factor that goes into both 186, 300. I really did sat here and pulled my hair out wondering what I was missing. As I said, I honestly did not see 2 x 3. I'm trying to figure this on my own since the exam monitors would likely not be happy if I brought someone into help with question. I appreciate the breakdown. I really do. (Whew) Moving away from fractions now and getting into "real number" issues/problems.

And BTW, if you have Skype, I have it too.

Yes, I can tell you are making a genuine effort to learn these concepts. And as Prove It mentioned above, you can also list the factors of both numbers and take the largest factor common to both as the GCD. For example:

300: 1, 2, 3, 5, 6, 10, 15, 20, 30, 50, 60, 100, 150, 300

186: 1, 2, 3, 6, 31, 62, 93, 186

And we see the largest factor (or divisor) common to both numbers is 6. Obviously, I prefer the rime factorization method since I have been pushing it. There is also the Euclidean algorithm:

$$300=1\times186+114$$

$$186=1\times114+72$$

$$114=1\times72+42$$

$$72=1\times42+30$$

$$42=1\times30+12$$

$$30=2\times12+6$$

$$12=2\times6+0$$

Since the last remainder is zero, the algorithm ends with 6 as the greatest common divisor of 300 and 186. This agrees with the GCD found by prime factorization. (Yes)