數學問題!!

1
Consecutive number must be one even and one odd,
so all the even number will not be the HCF.
with Consecutive number different = 1 and can not form HCF of all odd number
The HCF will be 1

In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. Some authors emphasize that "greatest" is not so much a measure of magnitude as a label for the fact that any other common divisor of two numbers divides their gcd. These authors also do not require the gcd to be positive, so that there are two gcds a and b, with a = −b.
The greatest common divisor of a and b is written as gcd(a, b), or sometimes simply as (a, b). For example, gcd(12, 18) = 6, gcd(−4, 14) = 2 and gcd(5, 0) = 5. Two numbers are called coprime or relatively prime if their greatest common divisor equals 1. For example, 9 and 28 are relatively prime.
The greatest common divisor is useful for reducing vulgar fractions to be in lowest terms. Consider for instance


圖片參考：http://upload.wikimedia.org/math/d/7/3/d7331627621926e10f3e8b874d1a25a8.png

where we cancelled 14, the greatest common divisor of 42 and 56.






Contents[hide]

1 Calculating the GCD
2 Properties
3 Probabilities and expected value
4 The gcd in commutative rings
5 See also
6 References
7 External links



[edit] Calculating the GCD
Greatest common divisors can in principle be computed by determining the prime factorizations of the two numbers and comparing factors, as in the following example: to compute gcd(18,84), we find the prime factorizations 18 = 2·32 and 84 = 22·3·7 and notice that the "overlap" of the two expressions is 2·3; so gcd(18,84) = 6. In practice, this method is only feasible for very small numbers; computing prime factorizations in general takes far too long.
A much more efficient method is the Euclidean algorithm: divide 84 by 18 to get a quotient of 4 and a remainder of 12. Then divide 18 by 12 to get a quotient of 1 and a remainder of 6. Then divide 12 by 6 to get a remainder of 0, which means that 6 is the gcd.
The series of quotients generated by the Euclidean algorithm comprise a continued fraction.

兩個連續數的H.C.F.是1。Beacuse it doesn't have H.C.F

兩個連續數的H.C.F.是1。

兩個連續數的H.C.F.就是1

兩個連續數的H.C.F.是1

1becausen,n+1 does not have H.C.F.

1 because n,n+1 does not have HCF.