 Euclidean algorithm

For positive integers $a,b$, the Euclidean algorithm repeatedly performs division with remainder: write $a=qb+r$ with $0\le r<b$, then replace $(a,b)$ by $(b,r)$ until the second entry is zero. The final nonzero entry is their greatest common divisor. A remainder preserves the set of common divisors: an integer divides $a$ and $b$ exactly when it divides $b$ and $a-qb$. The positive remainders strictly decrease, so the calculation terminates.

The complete calculation for $252$ and $198$ isHence $\gcd (252,198)=18$. The fraction $252/198$ reduces to $14/11$ by dividing numerator and denominator by that common factor. The same remainder calculation has a geometric interpretation: remove squares from a $252$-by-$198$ rectangle, then repeat on the remaining strip. Each division step records how many equal squares fit and the width of the new strip.

The procedure does more than test individual divisors. Each step replaces the whole common-divisor problem by a smaller equivalent one. Euclid's Book VII, Proposition 2 describes the corresponding repeated-subtraction construction; batching those subtractions into division with remainder gives the present formulation. ^[ref]