عندما يصبح الباقي صفرًا، فإن القاسم المشترك الأكبر هو آخر باقي غير صفري، وهو $15$.

Sun, 03 May 2026 09:44:39 +0000

When the Remainder is Zero, the GCD Is the Last Non-Zero Remainder — And It’s 15 In number theory, one of the most important principles when finding the greatest common divisor (GCD) of two integers is this: When the division process ends and the remainder becomes zero, the last non-zero remainder is always the GCD. This concept holds true regardless of the numbers involved — and in particular examples, it often leads to elegant results. Imagine performing the Euclidean algorithm to compute the GCD of two numbers. As you repeatedly divide and replace remainders, each step reduces the size of the numbers until the remainder reaches zero. At that moment, the prior remainder — the final non-zero value — is the greatest common divisor. This final non-zero remainder is not arbitrary; it’s the largest integer that divides both original numbers without leaving a remainder. For instance, consider two integers where the division process ends with a remainder of zero. Among all possible remainders during this algorithm, the last non-zero remainder is invariably the GCD. In this case, this final remainder equals 15.