Remainders and Euclid's Algorithm

The solution of the following problem includes an important idea:

Problem 5. Prove that the number n³+2n is divisible by 3for any natural number n.

Solution.The number n can give any of the following reminders when divided by 3: 0, 1 and 2.We consider the 3 cases.

If n has remainder 0, then both n³ and 2n are divisible by 3.

If n has remainder 1, then n³ has remainder 1, 2n has remainder 2 and 1+2 is divisible by 3.

If n has remainder 2, then n² has remainder 1 and n³ has remainder 2, 2n has remainder 1 and 1+2 is divisible by 3.

The Euclid's Algorithm gives us a less painful way to calculate the G.C.D. of larger numbers. The idea of the method is that any common divisor of two numbers a and b (a>b) also divides a-b; also, any common divisor of b and a-b divides the number a as well. Hence $\gcd(a,b)=\gcd(b, a-b)$

Problem 6. Find the greatest common divisor of 451 and 287. Solution:

$$\begin{eqnarray*}\gcd(451,287) & = & \gcd(287,164)\\ & = & \gcd(164,123)\\ &... ...41)\\ & = & \gcd(82,41)\\ & = & \gcd(41,41)\\ & = & 41.\\ \end{eqnarray*}$$

Methodological Remark. However simple it may seem, Euclid's algorithm is a very important arithmetic fact (which can be used, for example to prove the Fundamental Theorem of Arithmetic). Therefore, we think it would be wise a teacher to devote a separate session to this remarkable method (together with G.C.D and L.C.M., and their properties).