Find the Greatest Common Divisor of Two Integers

The Greatest Common Divisor (GCD) of two whole numbers, also called the Greatest Common Factor (GCF) and the Highest Common Factor (HCF), is the largest whole number that's a divisor (factor) of both of them. For instance, the largest number that divides into both 20 and 16 is 4. (Both 16 and 20 have larger factors, but no larger common factors -- for instance, 8 is a factor of 16, but it's not a factor of 20.) In grade school, most people are taught a "guess-and-check" method of finding the GCD. Instead, there is a simple and systematic way of doing this that always leads to the correct answer. The method is called "Euclid's algorithm." If you want to know how to truly find the Greatest Common Divisor of two integers, see Step 1 to get started.

Steps

Using the Divisor Algorithm

  1. Drop any negative signs.
  2. Know your vocabulary: when you divide 32 by 5,
      • 32 is the dividend
      • 5 is the divisor
      • 6 is the quotient
      • 2 is the remainder(or modulo).
  3. Identify the larger of the two numbers. That will be the dividend, and the smaller the divisor.
  4. Write out this algorithm: (dividend) = (divisor) * (quotient) + (remainder)
  5. Put the larger number in the spot for dividend, and the smaller number as the divisor.
  6. Decide how many times the smaller number will divide into the larger number, and drop it into the algorithm as the quotient.
  7. Calculate the remainder, and substitute it into the appropriate place in the algorithm.
  8. Write out the algorithm again, but this time A) use the old divisor as the new dividend and B) use the remainder as the new divisor.
  9. Repeat the previous step until the remainder is zero.
  10. The last divisor is the greatest common divisor.
  11. Here is an example, where we are trying to find the GCD of 108 and 30:
  12. Notice how the 30 and the 18 in the first line shift positions to create the second line. Then, the 18 and 12 shift to create the third line, and the 12 and 6 shift to create the fourth line. The 3, 1, 1, and 2 that follow the multiplication symbol do not reappear. They represent how many times the divisor goes into the dividend, so they are unique to each line.

Using Prime Factors

  1. Drop any negative signs.
  2. Find the prime factorization of the numbers, and list them out as shown.
    • Using 24 and 18 as the example numbers:
      • 24- 2 x 2 x 2 x 3
      • 18- 2 x 3 x 3
    • Using 50 and 35 as the example numbers:
      • 50- 2 x 5 x 5
      • 35- 5 x 7
  3. Identify all common prime factors.
    • Using 24 and 18 as the example numbers:
      • 24- 2 x 2 x 2 x 3
      • 18- 2 x 3 x 3
    • Using 50 and 35 as the example numbers:
      • 50- 2 x 5 x 5
      • 35- 5 x 7
  4. Multiply the common factors together.
    • In the case of 24 and 18, multiply 2 and 3 together to get 6. Six is the greatest common factor of 24 and 18.
    • In the case of 50 and 35, there is nothing to multiply. 5 is the only common factor, and therefore the greatest.
  5. Finished.

Tips

  • One way to write this, using the notation <dividend> mod <divisor> = the remainder is that GCD(a,b) = b if a mod b = 0, and GCD(a,b) = GCD(b, a mod b) otherwise.
  • As an example, let's find GCD(-77,91). First, use 77 instead of -77, so GCD(-77,91) becomes GCD(77,91). Now, 77 is less than 91, so we should swap them, but let's see how the algorithm takes care of that if we don't. When we calculate 77 mod 91, we get 77 (since 77 = 91 x 0 + 77). Since that's not zero, we switch (a, b) for (b, a mod b) and that gives us: GCD(77,91) = GCD(91,77). 91 mod 77 gives 14 (remember, that means 14 is the remainder). Since that's not zero, swap GCD(91,77) for GCD(77,14). 77 mod 14 gives 7 which is not zero, so swap GCD(77,14) for GCD(14,7). 14 mod 7 is zero, since 14 = 7 * 2 with no remainder, so we stop. And that means: GCD(-77,91) = 7.
  • This technique is very useful when reducing fractions. By the above example, the fraction -77/91 reduces to -11/13 because 7 is the greatest common divisor of -77 and 91.
  • If 'a' and 'b' are both zero, then any non-zero number divides them both, so there is technically no greatest common divisor in this case. Mathematicians often just say that the greatest common divisor of 0 and 0 is 0, and that's the answer this method gives.

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