2011 S1-02 Maths Blog: lcm-hcf

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Chapter 1 (Revision): LCM & HCF: Someone asked...

The HCF and LCM of two numbers are 12 and 5040 respectively. If one of the number is 420, what is the other number?
If the 2 numbers are A and B, rewriting what we are given:

HCF of A and B = 12 (which is the same as 2 x 2 x 3)
LCM of A and B = 5040

Now, to find HCF of A and B, we usually do prime factorisation of each number, then 'circle' the common factors.
In other words, with prime factorisation,

A = 2 x 2 x 3 x ? x ? x ... x ?
B = 2 x 2 x 3 x ? x ? x ... x ?
{where we do not know what are the factors and how many of them at this point}

Now, look at LCM... if we were to use long division method to find the LCM, it means....

2 ) A, B
2 ) A', B'
3 ) A'', B''
? ) A''', B''
................

So, it means LCM of these 2 numbers A and B = 2 x 2 x 3 x ? x ... x ? (but we have to bear in mind that 2, 2 and 3 are common for both numbers)
Given that one of the numbers is 420, we would be able to find the factors that this number contributes to the LCM.

Prime Factorisation of 420 = 2 x 2 x 3 x 5 x 7
Now, we know that LCM of A and B, 5040 is equal to 2 x 2 x 3 x 5 x 7 x ?
The remaining factor: 5040 ÷ (2 x 2 x 3 x 5 x 7) = 12
Hence the other number is a product of 2 x 3 x 3 x 12 = 144

Challenge: Shortcut to find LCM...

Jennifer thinks there is a shortcut to find out lowest common multiple (LCM) of 2 numbers.

She says that,
"If you multiple the two numbers, you will ALWAYS get the LCM."
Is Jennifer right?

Does her method work every time? Sometimes? Never?

Explain your reasoning.
Illustrate your reasoning with examples.

Source: http://ims.ode.state.oh.us/

Chapter 1 Activity: Factors and Multiples - Relationships of Numbers

Describe the relationship of these numbers: 12, 36 and 72.
Use words like "Factor", "Multiples", "Prime Numbers", "LCM", "HCF", etc to help describe the relationship.

The relationships are as follow: