Rivalry2

HTML Diff

0 added 0 removed

Original 2026-01-01

Modified 2026-02-28

1 The greatest common divisor (GCD) obeys certain mathematical properties. Here are a few important ones:

2 Commutative Property

3 Since the order of the numbers does not change the final result, GCD is commutative, i.e., GCD \( (a, b) = \) GCD\( (b, a)\)

4 For example: GCD\((12, 18) = \).GCD\((18, 12) = 6\)

5 Associative Property

6 The GCD is associative because the grouping of numbers does not change the result, i.e., GCD (\(a, \)GCD\((b, c)) =\) GCD(GCD\( (a, b), c)\)

7 For example,

8 <ul><li>First Grouping: GCD(\(8,\) GCD \((12,16))\)</li>

9 </ul>GCD\((12, 16) = 4\)

10 GCD \((8, 4) = 4\)

11 So, GCD(8, GCD \((12, 16)) = 4\)

12 <ul><li> Second Grouping: GCD(GCD\((8, 12), 16)\)</li>

13 </ul>GCD\((8, 12) = 4\)

14 GCD\((4, 16) = 4\)

15 So, GCD(GCD\((8, 12), 16) = 4\)

16 GCD\( (8,\) GCD\((12, 16)) =\) GCD(GCD\((8, 12), 16)\)

17 Hence, GCD is associative.

18 Distributive Property (Over LCM)

19 The GCD is distributive over the Least Common Multiple (LCM).

20 GCD\( (a, \)LCM\( (b, c)) = \)LCM (GCD \((a, b)\), GCD\((a, c)).\)

21 For example: GCD\( (8, \)LCM\( (12, 18)) = \)LCM (GCD \((8, 12),\) GCD \((8, 18))\)

22 Let’s look at the steps involved:

23 <ul><li>LCM \((12, 18) = 36 \) </li>

24 <li>GCD\( (8, 12) = 4\) </li>

25 <li>GCD\( (8, 18) = 2\) </li>

26 <li>LCM \((4, 2) = 4\) </li>

27 <li>GCD\( (8, 36) = 4\) </li>

28 </ul>Therefore,

29 GCD \((8,\) LCM\( (12, 18))\) = LCM (GCD\( (8, 12)\), GCD \((8, 18)) \)holds true.

30 Divisibility Property

31 According to the divisibility property, if d is given as GCD \((a, b),\) then it means d divides both a and b evenly. \(d ∣ a \) and \(d ∣ b\) where d = GCD (\(a, b)\).

32 Example: If d = GCD \((18, 24) = 6\), then \(6 ∣ 18\) and \( 6 ∣ 24\).

33 GCD with Zero

34 For any non-zero integer n, the GCD of 0 and n is n. However, the GCD of 0 and 0 is undefined because there is no greatest common divisor in this case.

35 GCD\((0, n) = n\), but GCD\((0, 0) =\) undefined.

36 Example: GCD\((12, 0) = 12\), but GCD\((0, 0)\) is undefined.

37 Multiplicative Property

38 According to the multiplicative property of GCD, if a and b are co-prime, then:

39 GCD\( (a × b, c) =\) GCD\( (a, c) ×\) GCD \((b, c). \)

40 For example: Consider a = 7, b = 5, and c = 20.

41 Since 7 and 5 are co-prime;

42 (GCD \((7,5) = 1)\), GCD \((7 × 5, 20) =\) GCD \((7,20) ×\) GCD \((5,20)\) \(= 1 × 5 = 5 \)

43 Hence, the property holds true.