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📝 Relatively Prime Numbers & Venn Diagrams
This page provides solutions and explanations for exercises related to G.C.F, L.C.M, and Venn diagrams, based on the Grade 6 curriculum.
1. Relatively prime: For two relatively prime numbers, which of the following statements are true? Identify all that apply.
a. Because there are no common factors in the intersection, the G.C.F is 0.
The Greatest Common Factor (G.C.F) is the largest number that divides two or more numbers. The smallest possible G.C.F for any pair of positive integers is 1. It can never be 0.
b. Because there are no common factors in the intersection, the G.C.F is 1.
This is the definition of relatively prime numbers. They have no common prime factors, so their only common factor is 1. In a Venn diagram, the intersection of their prime factors would be empty, meaning the G.C.F is 1.
c. The L.C.M is the product of the two numbers.
This is a key property of relatively prime numbers. Since they share no common factors, the Least Common Multiple (L.C.M) is simply found by multiplying the two numbers together. For example, the L.C.M of 8 and 15 (which are relatively prime) is \(8 \times 15 = 120\).
d. The L.C.M is the product of all prime factors in the Venn diagram.
This statement is true for finding the L.C.M of *any* two numbers using a Venn diagram, not just relatively prime numbers. You multiply all the prime factors listed in all parts of the diagram (left, right, and intersection).
2. Use Prime Factors: Answer each question.
a. How can the Venn diagram help you find the G.C.F?
The Venn diagram visually organizes the prime factors of two numbers.
To find the G.C.F (Greatest Common Factor), you look at the intersection (the overlapping part) of the two circles. The G.C.F is the product of all the prime factors found in this intersection. If the intersection is empty, the G.C.F is 1.
b. How can the Venn diagram help you find the L.C.M?
The Venn diagram contains all the unique prime factors of both numbers.
To find the L.C.M (Least Common Multiple), you multiply all the prime factors from all sections of the Venn diagram together (the numbers in the left part, the right part, and the intersection). This ensures every prime factor from both numbers is included exactly once in the product.