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Maximum common divisor (M.D.C.)


Two natural numbers always have common dividers. For example, the common dividers 12 and 18 are 1,2,3 and 6. Of these, 6 is the largest. So we call the 6 in maximum common divider of 12 and 18 and we indicate m.d.c. (12.18) = 6.

The largest common divisor of two or more numbers is called the maximum common divider of these numbers. We use the abbreviation m.d.c.

Some examples:
mdc (6.12) = 6
mdc (12.20) = 4
mdc (20.24) = 4
mdc (12.20.24) = 4
mdc (6.12.15) = 3

M.D.C. Calculation

A way to calculate m.d.c. of two or more numbers is to use the decomposition of these numbers into prime factors.

1) we break down the numbers into prime factors;
2) the m.d.c. is the product of common prime factors.

Follow the calculation of m.d.c. between 36 and 90:
36 = 2x 2 x 3 x 3
90 = 2 x 3 x 3 x 5

The m.d.c. is the product of common prime factors => m.d.c. (36,90) = 2x3x3
Therefore m.d.c. (36.90) = 18.

Writing the factorization of number in the form of power we have:
36 = 22 x 32
90 = 2 x 32 x5
So m.d.c. (36.90) = 2 x 32 = 18.

O m.d.c. of two or more numbers, when factored, is the product of the factors common to them, each raised to the lowest exponent.

M.D.C. Calculation by the process of successive divisions

In this process we make several divisions until we reach an exact division. The divisor of this division is m.d.c. Follow the calculation of m.d.c. (48,30).

Practical rule:

1º) we divide the larger number by the smaller number;
48 / 30 = 1 (with remainder 18)

2º) we divide divisor 30, which is divisor of the previous division, by 18, which is the rest of the previous division, and so on;
30 / 18 = 1 (with remainder 12)

18 / 12 = 1 (with remainder 6)

12 / 6 = 2 (with zero remainder - exact division)

3º) O exact division divider it's 6. So m.d.c. (48.30) = 6.

Prime numbers among themselves

Two or more numbers are cousins ​​among themselves when the maximum
common divisor of these numbers is 1.

Examples:
The numbers 35 and 24 are prime numbers to each other, since mdc (35,24) = 1.
The numbers 35 and 21 they are not prime numbers to each other, since mdc (35,21) = 7.

Property of M.D.C.

Among the numbers 6, 18 and 30, the number 6 is a divisor of the other two. In this case, 6 is m.d.c. (6,18,30). Watch:

6 = 2x3
18 = 2x32
30 = 2x3x5
So m.d.c. (6.18,30) = 6

Given two or more numbers, if one of them is divisor of all the others, then he is m.d.c. of the given numbers. Next: Least Common Multiple (M.M.C.)