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18 Dec 2012, 03:40
Kudos
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I came across these concept of number property ,which i read but still not able to understood properly...as WHY and HOW can i implement those concepts
The concept which i read is from MGMAT number property chapter 10:
1) Consecutive multiples of 'n' have a G.C.F of 'n'
2) The G.C.F of two numbers cannot be larger than difference between two number.
Why these concepts are formulated in these way and How to implement on the questions?
Thanks in advance.
Regards,
Aristocrat
The concept which i read is from MGMAT number property chapter 10:
1) Consecutive multiples of 'n' have a G.C.F of 'n'
2) The G.C.F of two numbers cannot be larger than difference between two number.
Why these concepts are formulated in these way and How to implement on the questions?
Thanks in advance.
Regards,
Aristocrat
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18 Dec 2012, 04:00
Kudos
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Aristocrat
Consecutive multiples of 'n' have a G.C.F of 'n'
This implies that the greatest common factor of \(nk\) (multiple of n) and \(n(k+1)\) (next multiple of n) is \(n\). Well, this must be true since \(k\) and \(k+1\) are consecutive integers, thus they don't share any common factor but 1 (for example two consecutive integers 15 and 16 do not share any common factor but 1). Therefore the GCF of \(nk\) and \(n(k+1)\) is \(n\).
For example, consider consecutive multiples of 7: 70 and 77 --> GCF(70, 77)=7.
The G.C.F of two numbers cannot be larger than difference between two number.
This should be formulated as:
For two distinct positive integers \(a\) and \(b\) (\(a>b\)): \(GCF(a,b)\leq{a-b}\), the greatest common divisor of two distinct positive integers cannot be greater than their positive difference. Proof: any common factor of two integers is also a factor of their sum and difference, hence GCF of two distinct integers cannot be greater than the positive difference between them as in this case GCF must also be a factor of the difference which is less then it, which is impossible.
For example if \(a=25\) and \(b=20\) then the greatest common divisor of 25 and 20 can not be more than 25-20=5 (as number more than 5 cannot be divisor of 5, which is the difference between 25 and 20).
Hope it's clear.
Aristocrat
In Quant, you can establish innumerable inferences from the theory of any topic. The point is that you should be comfortable with the theory. If I give you a statement, you should be able to say whether it is true or false based on your conceptual understanding. There is no point memorizing these facts. Just try to understand why the book says it is so. Next time, if you come across a situation dealing with GCF, you don't need to 'recall' these; you will know that these are true.
1) Consecutive multiples of 'n' have a G.C.F of 'n'
What are consecutive multiples?
e.g. 4n, 5n or 18n, 19n etc are pairs of consecutive multiples of n. What will be the greatest common factor of 18n and 19n? We know that n is their common factor. Is there any common factor between 18 and 19 (except 1)? No. So GCF will be n only. Take any two consecutive numbers. They will have no common factor except 1. Hence, if we have two consecutive factors of n, their GCF will always be n.
2) The G.C.F of two numbers cannot be larger than difference between two numbers.
This is true in case the numbers are distinct. GCF is a factor of both the numbers. Say GCF of two distinct numbers is x. This means the two numbers are mx and nx where n and m have no common factor. What can be the smallest difference between m and n? m and n can be consecutive numbers. In this case, the difference between nx and mx will be x which is equal to the GCF. If m and n are not consecutive, the difference between mx and nx will be much larger than x. The difference between mx and nx cannot be less than x.
Say, GCF of two numbers is 6. The numbers can be 6 and 12 or 6 and 36 etc but they cannot be 6 and 8 since both numbers must have 6 as a factor.
