pure math HCF

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.
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, which uses the division algorithm in combination with the observation that the gcd of two numbers also divides their difference: 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 compose a continued fraction.
http://en.wikipedia.org/wiki/Highest_common_factor
I hope I can help.
祝學業進步.

Highest Common Factors can in principle be computed by determining the prime factorizations of the two numbers and comparing factors, as in the following example:
To compute hcf (18,84),
we find the prime factorizations 18 = 2•32 and 84 = 22•3•7 and
we notice that the "overlap" of the two expressions is 2•3;
so hcf (18,84) = 6.