Prime factorizations

Complete the following tasks.

1. Create a Venn diagram using the prime factorization of 48 in one set and the prime factorization of 72 in the other set.
2. The LCM and GCF are given. Explain how you could use the Venn diagram to obtain the same answers.
   • LCM(48, 72) = 144
   • GCF(48, 72) = 24
3. Next, we will explore a conjecture about LCM and GCF.
   1. What is the product of 12 and 100?
   2. What is the product of LCM(12, 100) and GCF(12, 100)?
   3. Provide a conjecture based on the previous results.
   4. Use prime factorizations to justify why the conjecture might be true.

Challenge problems

1. Will the LCM GCF conjecture work for three numbers? Pick three numbers to see.
2. Construct three numbers, each with exactly seven factors.
3. How many two-digit numbers have 8 factors? Justify your answer. (difficult question!)

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