Cambridge101
New member
- Joined
- Dec 16, 2021
- Messages
- 49
Hi there,
Just a question I would like fully answered. Often students are taught just to accept that expressing a positive integer as a product of its primes will mean you can find the HCF and the LCM, there is never really an explanation for why this is the case. Could someone explain?
Intuitively, I can see that when one larger integer is a multiple of the smaller integer - the smaller integer must be the HCF. Simply by knowing that the highest factor of an integer is always itself. Thus, if the larger integer is a multiple of the smaller number, by definition this means we can produce this larger integer, by multiplying the smaller integer by some other integer. Thus the smaller integer is a factor of the larger integer. Meaning it is the HCF for the pair of integers, because the smaller is the highest factor for itself. Likewise, it is intuitive to see that when we have two integers and one is a multiple of the other, the LCM is always the bigger integer. Because every numbers lowest multiple is itself. So if the larger integer is a multiple of the other and itself. This by definition must be the LCM.
But why does expressing an integer using only a combination of prime factor help us find the HCF and the LCM for other numbers.
Just a question I would like fully answered. Often students are taught just to accept that expressing a positive integer as a product of its primes will mean you can find the HCF and the LCM, there is never really an explanation for why this is the case. Could someone explain?
Intuitively, I can see that when one larger integer is a multiple of the smaller integer - the smaller integer must be the HCF. Simply by knowing that the highest factor of an integer is always itself. Thus, if the larger integer is a multiple of the smaller number, by definition this means we can produce this larger integer, by multiplying the smaller integer by some other integer. Thus the smaller integer is a factor of the larger integer. Meaning it is the HCF for the pair of integers, because the smaller is the highest factor for itself. Likewise, it is intuitive to see that when we have two integers and one is a multiple of the other, the LCM is always the bigger integer. Because every numbers lowest multiple is itself. So if the larger integer is a multiple of the other and itself. This by definition must be the LCM.
But why does expressing an integer using only a combination of prime factor help us find the HCF and the LCM for other numbers.