Hi all. So Im feeling totally lost on the GCF/LCM/LCD type questions. I get the concept when we use real numbers, but a lof of the times i get stumped on the DS questions involving these values.
Can anyone share a quick primer on this, specific to GMAT, and share the concepts ?
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- StaceyKoprince
- ManhattanGMAT Staff
- Posts: 9360
- Joined: Wed Oct 19, 2005 9:05 am
- Location: Montreal
greatest common FACTOR and least common MULTIPLE
Notice the words I capitalized - that should be your focus when you think about each concept. DO NOT think about "greatest" and "least" even though those are the first words; that's what messes everybody up.
Factors are smaller than or equal to the main number
eg, 1, 2, 4, and 8 are all factors of 8
Multiples are greater than or equal to the main number
eg, 8, 16, 24, and 32 are all multiples of 8
I'll show you the theoretical way of handling GCF and LCM, though if the numbers are small enough, you can also just try real numbers. But I assume you don't struggle with real numbers as much as with theory.
1) break your numbers down to their prime factors.
eg, 8 = 2*2*2 and 10 = 2*3
2) Write everything in terms of exponents:
8 = 2^3
10 = 2^1 * 5^1
3) For GCF, we want a FACTOR, so we want a SMALLER number than our starting point. Choose the smallest EXPONENT for each prime base
eg 2 is part of both 8 and 10. The smaller exponent is 1 (for 2^1, which is part of 10). Choose that.
Dealing with 5 is a little trickier because 8 doesn't have any 5's. That's the equivalent of saying 8 = 2^3 * 5^0 (remember that anything to the zero power is 1). So if you have 5^0 and 5^1, which is the smaller exponent? Zero, of course, so choose 5^0 - which means we can just ignore this because 5^0 = 1.
So, GCF = 2
4) For LCM, we want a MULTIPLE, so we want a LARGER number than our starting point. Choose the largest EXPONENTS.
eg for 2, the largest exponent is 3, so select 2^3. For 5, the largest exponent is 1, so select 5^1. Multiply these to get 2^3 * 5^1 = 8*5 = 40.
Try with some harder numbers. Find the LCM and GCF of 18 and 60. (Try this before you look at the solution below!!)
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18 = 2*3*3 = 2^1 * 3^2
60 = 2*2*3*5 = 2^2 * 3^1 * 5^1
18 doesn't contain any 5's, so I'm going to rewrite it as: 2^1 * 3^2 * 5^0
GCF - choose smaller exponents
2^1
3^1
5^0
multiply to get 6
LCM - choose larger exponents
2^2
3^2
5^1
multiply to get 180
Notice the words I capitalized - that should be your focus when you think about each concept. DO NOT think about "greatest" and "least" even though those are the first words; that's what messes everybody up.
Factors are smaller than or equal to the main number
eg, 1, 2, 4, and 8 are all factors of 8
Multiples are greater than or equal to the main number
eg, 8, 16, 24, and 32 are all multiples of 8
I'll show you the theoretical way of handling GCF and LCM, though if the numbers are small enough, you can also just try real numbers. But I assume you don't struggle with real numbers as much as with theory.
1) break your numbers down to their prime factors.
eg, 8 = 2*2*2 and 10 = 2*3
2) Write everything in terms of exponents:
8 = 2^3
10 = 2^1 * 5^1
3) For GCF, we want a FACTOR, so we want a SMALLER number than our starting point. Choose the smallest EXPONENT for each prime base
eg 2 is part of both 8 and 10. The smaller exponent is 1 (for 2^1, which is part of 10). Choose that.
Dealing with 5 is a little trickier because 8 doesn't have any 5's. That's the equivalent of saying 8 = 2^3 * 5^0 (remember that anything to the zero power is 1). So if you have 5^0 and 5^1, which is the smaller exponent? Zero, of course, so choose 5^0 - which means we can just ignore this because 5^0 = 1.
So, GCF = 2
4) For LCM, we want a MULTIPLE, so we want a LARGER number than our starting point. Choose the largest EXPONENTS.
eg for 2, the largest exponent is 3, so select 2^3. For 5, the largest exponent is 1, so select 5^1. Multiply these to get 2^3 * 5^1 = 8*5 = 40.
Try with some harder numbers. Find the LCM and GCF of 18 and 60. (Try this before you look at the solution below!!)
-
-
-
-
-
18 = 2*3*3 = 2^1 * 3^2
60 = 2*2*3*5 = 2^2 * 3^1 * 5^1
18 doesn't contain any 5's, so I'm going to rewrite it as: 2^1 * 3^2 * 5^0
GCF - choose smaller exponents
2^1
3^1
5^0
multiply to get 6
LCM - choose larger exponents
2^2
3^2
5^1
multiply to get 180
Stacey Koprince
Instructor
Director, Content & Curriculum
ManhattanPrep
Instructor
Director, Content & Curriculum
ManhattanPrep
- rfernandez
- Course Students
- Posts: 381
- Joined: Fri Apr 07, 2006 8:25 am
- StaceyKoprince
- ManhattanGMAT Staff
- Posts: 9360
- Joined: Wed Oct 19, 2005 9:05 am
- Location: Montreal
8 posts Page 1 of 1