Trick with finding prime factorization
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[Originally posted in Discussions]
You see, there is a peculiar pattern with some numbers. I find that if I cannot figure out the prime factorization of an even number, if you multiply the first number by the second number (both have to be even), and divide it by two, it nearly always comes out as the prime factorization. I am not really sure why, but that is what happens. For example, on this problem, if you multiply 360 by 100 and divide by two, you get 1,800. That was the answer. Could someone tell me why this happens?
That's an interesting hypothesis you have! Unfortunately, it seems like you've made a multiplication mistake. 360100/2 = 18,000 but lcm(360, 100) = 1,800. They are off by a factor of 10. This property also doesn't hold true for all even numbers. Take 8 and 4 for instance. lcm(8, 4) = 8, but 84/2 = 16. However, you could ask yourself some follow up questions to keep exploring:
Why do 360100/2 and lcm(360, 100) differ by a factor of 10? What is special "how many 10s" are in a number (referring to the prime factorization of 10)? Challenge problem: how many zeroes does 98765432*1 end in?
What numbers does this property hold true for? If you have a and b, when does a*b/2 = lcm(a, b)?
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