If p and q are relatively prime, then GCF(p,q) = ______, and LCM(p,q) = _______.
GCF=1
LCM=pq
If p and q are relatively prime, it means that their greatest common factor (GCF) is 1. So, GCF(p, q) = 1.
The least common multiple (LCM) of two numbers is the smallest positive integer that is divisible by both numbers. In this case, since p and q are relatively prime, there are no common factors other than 1. Therefore, the LCM(p, q) would simply be the product of p and q.
So, LCM(p, q) = p * q.
If p and q are relatively prime, it means that there are no common factors other than 1 between them. In this case, the greatest common factor (GCF) of p and q is 1, and the least common multiple (LCM) of p and q is simply their product, pq.
To understand why this is the case, let's first define GCF and LCM:
GCF (Greatest Common Factor): The GCF of two numbers is the largest positive integer that divides both of them without leaving a remainder.
LCM (Least Common Multiple): The LCM of two numbers is the smallest positive integer that is divisible by both of them.
To find the GCF and LCM of two numbers, we need to factorize them into their prime factors.
Let's take an example with p = 9 and q = 14:
Step 1: Factorize 9 and 14
9 = 3^2 (prime factorization of 9: 3 × 3)
14 = 2 × 7 (prime factorization of 14)
Step 2: GCF
To find the GCF, we look for the common factors in both factorizations. In this case, there are no common factors other than 1, which means the GCF is 1.
Step 3: LCM
To find the LCM, we include all the prime factors from both factorizations, taking the highest power of each. In this case, we have 2, 3^2, and 7. Multiplying them together, we get:
LCM = 2 × 3^2 × 7 = 2 × 9 × 7 = 126
So, in this example, GCF(9, 14) = 1, and LCM(9, 14) = 126.
In general, if p and q are relatively prime, their GCF is always 1, and their LCM is always equal to the product of p and q.
gcf=1
lcm=p,q