given the lcm and hcf

reversing the question
given the lcm and hcf of a pair of numbers
what could they be?

the powerpoint is here

from an idea by David Wells (question 1)

lcm and hcf generalising

deducing the hcf and lcm of two numbers from their prime factorisations
the powerpoint is here

type 1

type 2

products of primes

the powerpoint is here
[ it needs to be downloaded for the animations to work ]

what is the common form of these pairs of numbers?

what, in general, will the hcf be?

the lcm?

sequences of multiples

HCF

looking at the hcf (sometimes called the greatest common divisor) of two numbers
student: "Why didn't you tell us that you can just subtract the two numbers to get the hcf?"
teacher: "Because it doesn't always work"
student: "It does on the two examples we've looked at"
teacher: "Oh, so how about 5 and 15?"
student: "10"
teacher: "It didn't work"
student: "Well when does it work then?"

HCF and LCM problems

based on a KS2 SAT question
















the following problems are slightly adapted from Ms Sia's Realm of Maths blog
many thanks for these

highest common factor

this method, of lining up the factors in common, seems a helpful way to try to find the highest common factor of two numbers
having already done a prime factor break down

for example, for 1050 and 1540:

1050 = 2 x 3 x 5 x 5 x 7

1540 = 2 x 2 x 5 x 7 x 11,

putting loops around the common factors shows more clearly those numbers that are in both sets:

the product of these is the hcf

the product of what is left and the hcf is the lcm

the lcm can be calculated from the hcf since  hcf (a,b) x lcm (a,b) = a x b
so to find the lcm (often the trickier task) you can divide the product of the two numbers by their hcf

why is this relationship true?


Jo Morgan, at resourceaholic, suggests using a division by a common factor method:

a Venn diagram can be helpful

place the common factors in the intersection of two sets
hcf = product of the elements in (A intersection B)

the lowest common multiple is the product of all the elements in (A union B)

e.g. for 48 and 180

hcf is 2 x 2 x 3 = 12

lcm is 12 x 4 x 15 = 720








then, of course, there is always Euclid's algorithm
e.g. the highest common factor of 48 and 180 is
also the highest common factor of 180 - 48
and 180 - 2 x 48
and 180 - 3 x 48 = 36
so the problem is simplified to finding the hcf of  36 and 48