Tuesday, January 19, 2016

[S1_Expository] Exploring the HCF and LCM with a calculator

How does it work?
When we reduce a fraction to its lowest terms, we actually cancel out as many common factors as possible.  So eventually the numerator and the denominator of original fraction get cancelled by their Highest Common Factor (HCF), a.k.a. “Greatest Common Divisor (GCD) ” in America.  Just as England and the United States are divided by a common language(*), we can divide  756  by  6  to get the HCF.  Why?  That is because  756  was divided by the HCF to get  6.  You may try a similar trick with the denominators.  You get the same conclusion.
It is useful to know that the product of two numbers is equal to the product of their  HCF  and  LCM.  So   756 × 1386 = HCF × LCM.  Hence we have
Observe that the bracketed number  (756/126)  is equal to  6  which is the numerator of the reduced fraction.  So we do not even need to make that calculation.  Just take the reduced numerator  6  and multiply that with the original denominator  1386  to get the LCM.  You may try a similar trick with  1386  and  126.  You end up multiplying  756  by  11,  which gives the same answer.

Further Exploration
Try the above with different pairs of numbers.  How can you extend this to find the HCF and LCM of three or more numbers?

(*) OK, just kidding.  In Singapore, we sort of follow British English, but we are flexible.

Suitable Levels
Lower Secondary Mathematics (Sec 1 ~ grade 7)
* other syllabuses that involve factors, HCF (GCD) and LCM
* any interested learner