Greatest Common Multiple vs. Greatest Common Factor

**Anandha** · 10-31-2005, 06:14 AM

Friends,

Are these two terms the same? What is the difference and most importantly how to find them?

Please calrify this. I am stuck whenever I get questions on these two terms.

Thank you.

**Macintosh** · 10-31-2005, 07:22 PM

are you asking about the least common multiple?, then

for numbers 12 and 21, the least common multiple is 84. First, you break the number down into prime factors: 12- 2, 2, 3; and 21- 3, 7. Then, look for factors that comprise both numbers, discarding the extras. Since we have 2, 2, 3, 3, 7, we take 2, 2, 3, and 7, thus the LCM is 84. Sometimes, the books tell us to take the largest number, in this case, 21 and keep multiplying by 2, 3, 4, etc until you find a number that is divisible by the smaller number 12.

Any corrections are welcome!

**Anandha** · 11-01-2005, 08:16 AM

Machintosh,

I am looking for GREATEST COMMON FACTOR / MULTIPLE / DIVISOR

I know LCM, but don't know any of the above.

Please help. Thanks.

**800Bob** · 11-01-2005, 08:55 AM

Greatest Common Factor and Greatest Common Divisor are the same thing.

There is no such thing as a "Greatest Common Multiple." For any common multiple you can find, there will always be a greater one. For example, the common multiples of 8 and 12 are 24, 48, 72, 96, 120, etc. There is no greatest one.

**Anandha** · 11-02-2005, 11:13 PM

800Bob,

How to find greatest Common Factor / Divisor?

Thank you for clarifying.

**800Bob** · 11-03-2005, 10:10 AM

Originally Posted by Anandha

How to find greatest Common Factor / Divisor?

Express the integers as products of prime numbers, and then take the intersection of the prime factorizations.

Example:
To find the GCF of 144 and 168, first express both integers as products of primes:
144 = 2*2*2*2*3*3
168 = 2*2*2*3*7

Now take the intersection of these factorizations:
GCF = 2*2*2*3 = 24

**Anandha** · 11-04-2005, 08:25 AM

800Bob, got it!

Thank you very much!

**Bright** · 11-23-2005, 06:23 AM

800bob,

Is there any quick way to find the least common multiple? For ex. 144 and 168??

**800Bob** · 11-23-2005, 09:39 AM

Originally Posted by Bright

800bob,

Is there any quick way to find the least common multiple? For ex. 144 and 168?

Yes. Express both numbers as the product of prime numbers and take the larger power of each prime factor.

144 = 2^4 * 3^2
168 = 2^3 * 3 * 7

The larger power of 2 is 2^4. The larger power of 3 is 3^2. And the larger power of 7 is 7. So:

LCM = 2^4 * 3^2 * 7 = 1008