Relatively Prime Numbers.

Two numbers are called relatively prime or co-prime if they have no common divisors greater than 1.

The following facts are easy to prove and often used:

1.

If some natural number is divisible by two relatively prime numbers p and q, then it is divisible by their product pq.

2.

If the number pA is divisible by q, where p and q are relatively prime, then A is also divisible by q.

We introduce two more important definitions:

1.

The Greatest Common Divisor $\gcd(x,y)$ of two natural numbers x and y, and

2.

The Least Common Multiple lcm(x,y)

Here are some exercises on these:

Problem 4. Find gcm(A,B) of the numbers $A=2^3\cdot3^{10}\cdot5\cdot7$ and $B=2^5\cdot3\cdot11$ and lcm(C,D) of the numbers $C=2^8\cdot5^3\cdot7$ and $D=2^5\cdot3\cdot5^7$.

Answer. $\gcd(A,B)=24=2^3\cdot3$, and $\mathrm lcm(C,D)=420,000,000=2^8\cdot3\cdot5^7\cdot7$.