Number of Factors

I have been thinking about Prime Factors again and more precisely how many factors a number has. I was thinking about the smallest number with just 1 factor and then the smallest with 2 factors and so on. I liked the reminder that the odd number of factors belonged to square numbers. There was so much to notice as I complied a list of the first 12 smallest numbers that had just 1 factor, 2 factors etc. I liked thinking of the combinations that give us the factors of a number. I liked thinking about the fact that 60 has 12 factors because 3×2×2=12 (1 more than 2 × by 1 more than 1 × by 1 more than 1) and that 25 × 3 is also a number with 12 factors. The issue I have now, is which slide below will I show my class when I use it… do I want them to write each number as a product of prime factors and then notice the number of factors are increasing? or start with the number of factors and ask them the to find the smallest number that it works for or do I give them a mix of all different columns filled in?

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