The HCF of five unequal numbers is 430. Then 430 must necessarily be (a) Smallest of the three numbers (b) Multiple of the smallest number (c) Either of the above (d) None of these

Answer 1

Hi Rajnish G.,

Not a tough one! Your terms to keep in mind are: anything and everything.

Highest Common Factor (HCF) means, it is a factor (=will evenly divide) each of the five numbers. So when you have a choice that talks about "the three numbers", you should go "What??"

If it is a factor, it can't be GREATER than the number it is a factor of!

So, neither (a) nor (b). Does that give you a hint?

P.S. It may also be called the Greatest Common Factor (GCF).

You find it by prime-factoring each of your input numbers, then writing down (as a string list) only factors which appear in the factor lists of all your input numbers. If a prime number appears more than once in all the factor lists, then list it that number of times in your HCF string. When you've looked at all your factor sets (you can stop when you get to the lowest prime on any of the input number factor lists -- why?), then multiply the numbers in your string together. The product is your HCF.(the everything relation -- it had to "appear" in everything!)

The other term you'll see in these problems concerning factoring and sets, is "Least Common Multiple" (LCM). To find that one, factor each of the input numbers into its prime factors, and list these factors (suggestion: start by trying 2 as a factor, until it won't divide evenly any more; then move on to trying 3, and so on. Eventually you will have divided your original input number down to 1. So then move on to the next input number, and repeat the process.). If some number is a prime factor more than once, write it as such (more than once) in your list of factors. Then pick and choose factors for your LCM: for each integer that is a factor of any of your input numbers, scan through your lists and select the maximum number of times it appears. Write it that number of times. Do this for all integers up through the highest prime factor you found. Multiply your list of pick-and-choose factors together: the product is your LCM. (the anything relation -- if it appeared in anything, it was collected!)

Now, what use are HCF and LCM in the real world (other than making primary-school students puzzle over the idea of "multiplication families")? Not too much, other than giving you practice in mental math, which is not wasted time, you are growing your brain! You could use the idea of HCF as a block to efficiently tile a rectangular area of the edge sizes of the stated numbers (or do the same for a higher dimensional equivalent), and actually there is some neat geometry-art combination possible there, too. (And don't confine your attention to rectangular shapes, irregular plane tilings are way cool.) You may enjoy (if you're artistic) trying to work out what the equivalent use of the LCM might be (hint: don't be afraid to slice away!). And if that really gets your mental juices going, feel free to scramble out onto the web to find out the latest on slicing and reassembling shapes into other shapes -- it's actually an area at the forefront of mathematical research, check out: https://www.quantamagazine.org/mathematicians-cut-apart-shapes-to-find-pieces-of-equations-20191031/ . Happy mathing! -- Cheers, Mr. d.