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I'm following along in a math book I'm reading and the task at hand is to find the HCF of $270$ and $900$ using prime factorization. I know the answer is $90$ because I checked the answer at the back of the book and got it wrong.

I know that the only prime factors that go into each of them are $2, 3$ and $5.$ However I'm at a complete loss figuring out where to go from there to get $90.$

PNT
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General raid
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$$270 = 2\cdot 5\cdot3^3,\quad 900 = 2^2\cdot 5^2 \cdot 3^2.$$ The HCF is found by taking the smallest exponent of each distinct prime in the products. So, $$HCF(270,900) = 2\cdot 5\cdot 3^2 = 90.$$

Doug
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  • When you say exponent do you mean power? If yes, how do you determine which prime the exponent goes to and to what multiple, as in 3 to the power of two or three to the power of 3? I got it right at one stage but it was just from trial and error. – General raid Jun 27 '22 at 20:32
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    I'm picking the factor with the smaller power each time. $2^1$ vs. $2^2$, $3^2$ vs $3^3$... Make sense? – Doug Jun 27 '22 at 20:33
  • Sorry I'm not really following, let's say I have the list of common primes, in this instance 2, 3 and 5. What method do I need to follow to determine which way to multiply and exponentiate them in order to get the HCF? – General raid Jun 27 '22 at 20:55
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    Let's say the number $2$ is common to both prime factorisations. Look at each number's prime factorisation. Choose the power of $2$ that is the larger of the two. This becomes one factor of the HCF. Proceed to the next distinct prime... – Doug Jun 27 '22 at 20:58
  • Give yourself a change to practice: Find the HCF of 45 and 250. I will check your answer. – Doug Jun 27 '22 at 21:00
  • Ok so the prime factors of 45 and 250 are 3 and 5, by guesswork I figured 3 X 3 X 5 = 45, but it was simply that, guesswork and trial and error. I can't figure out what to do after that with 250 either. – General raid Jun 27 '22 at 21:17
  • Oh, I see. You are saying you have to guess to find prime factorisation? Here is the method I use: Find smallest prime factor first. Then, divide by that number as many times as you can. When you can no longer divide, move onto the next prime number and repeat. – Doug Jun 27 '22 at 21:20
  • I actually made a mistake there the prime factorisation of 45 is 3 and 5 whereas 250 is 2 and 5. So here's what I calculated by trial and error... 3 X 3 X 5 is 45 and 2 X 5 X 5 X 5 is 250 – General raid Jun 27 '22 at 21:24
  • Let us continue this discussion in chat. – General raid Jun 27 '22 at 21:32
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