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Secondary 1 | Maths
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Question involving HCF and LCM.
I've worked out this so far but I'm not sure what to do next? I looked at the textbook's example but it's just confusing. If you could, please explain your answer. Thank you for the help. :)
Date Posted: 3 years ago
Views: 660
Both numbers are greater than their HCF.
So all their common factors consist of 5² and 7 only.
Each number also has some other factors that the other doesn't have.
Now, the LCM has a 2³ and a 3².
Since they have no other common factors other than 5² and 7, then this means 1 number has 2³ and not 3², and likewise, the other number has 3² and not 2³.
This is why the LCM has to have both 3² and 2³ such that it is the smallest possible number that can be divided by both numbers exactly.
(If they had least one 3 or 2 each, then 2 and 3 are also common factors , which means the HCF would be higher than 175 already)
So the numbers are :
175 x 2³ = 175 x 8 = 1400
And
175 x 3² = 175 x 9 = 1575
So all their common factors consist of 5² and 7 only.
Each number also has some other factors that the other doesn't have.
Now, the LCM has a 2³ and a 3².
Since they have no other common factors other than 5² and 7, then this means 1 number has 2³ and not 3², and likewise, the other number has 3² and not 2³.
This is why the LCM has to have both 3² and 2³ such that it is the smallest possible number that can be divided by both numbers exactly.
(If they had least one 3 or 2 each, then 2 and 3 are also common factors , which means the HCF would be higher than 175 already)
So the numbers are :
175 x 2³ = 175 x 8 = 1400
And
175 x 3² = 175 x 9 = 1575
you can think of HCF as the overlap between the prime factors of both numbers,
and the LCM as a set of all the prime factors from both numbers, but without double counting the overlap.
Eg.
36 = 2 x 3 x 3 x 2 = 2² x 3²
150 = 5 x 5 x 3 x 2 = 5² x 3 x 2
Both numbers have one 2 and one 3 in common ('overlapping').
HCF = 3 x 2 = 6
A set of prime factors that covers both numbers without double counting the overlap would be :
[2,3,(3,2],5,5)
The square brackets cover all the prime factors of 36 and the curved brackets cover all the prime factors of 150.
In between, we can see that they have one 2 and 3 common. This is the overlap (HCF)
The LCM would be the entire set , but not double counting the overlap.
LCM = 2 x 2 x 3 x 3 x 5 x 5
= 2² x 3² x 5²
= 4 x 9 x 25
= 900
and the LCM as a set of all the prime factors from both numbers, but without double counting the overlap.
Eg.
36 = 2 x 3 x 3 x 2 = 2² x 3²
150 = 5 x 5 x 3 x 2 = 5² x 3 x 2
Both numbers have one 2 and one 3 in common ('overlapping').
HCF = 3 x 2 = 6
A set of prime factors that covers both numbers without double counting the overlap would be :
[2,3,(3,2],5,5)
The square brackets cover all the prime factors of 36 and the curved brackets cover all the prime factors of 150.
In between, we can see that they have one 2 and 3 common. This is the overlap (HCF)
The LCM would be the entire set , but not double counting the overlap.
LCM = 2 x 2 x 3 x 3 x 5 x 5
= 2² x 3² x 5²
= 4 x 9 x 25
= 900
This is analogous to those P6 area questions, where a figure consists of two overlapping shapes. (eg. Squares, triangles circles)
The HCF is akin to the overlapping area, and the LCM is akin to the total area of the figure.
Another thing to remember :
HCF x LCM = product of both numbers.
If we list out all the prime factors and take out the LCM, we are left with the HCF.
This is like the area of figure , whereby combined area of both shapes = area of figure + area of overlap
The HCF is akin to the overlapping area, and the LCM is akin to the total area of the figure.
Another thing to remember :
HCF x LCM = product of both numbers.
If we list out all the prime factors and take out the LCM, we are left with the HCF.
This is like the area of figure , whereby combined area of both shapes = area of figure + area of overlap
Wow, that was explained really well. Thanks for the additional info too.
Didn't know about that one :)
Didn't know about that one :)
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See main comments section for explanation.
Thank for the helpful explanation! It worked wonders
No prob
done
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Hope this helps:)
Date Posted:
3 years ago
It definitely did, thank you so much!
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