The product of two numbers is 2189 and their HCF is 12 .Find the LCM
Answers
Answer:
Interesting Number Theory fact: the product of two whole numbers is equal to the product of their HCF and LCM. To see why this is so, let H = the highest common factor of the two numbers a and b, and let L = their least common multiple.
There are two whole numbers p and q with a = Hp and b = Hq; in other words p and q contain the factors of a and b, respectively, that they do not share. Note that p and q do not have any common factors between them, because if they did, those factors would be common factors of a and b and would be accounted for in H. (At this point, for the sake of clarity, I am going to forego mathematical precision.) This means that q is the number to multiply a by to get L, the LCM, and p is the number to multiply b by to get L. (This can all be made exact and proper with cumbersome notation.) Carry out those multiplications:
Multiply both sides of a = Hp by q to get aq = Hpq = L.
Multiply both sides of b = Hq by p to get bp = Hpq = L.
Now consider the product abpq:
abpq = (aq)(bp) = (Hpq)(Hpq) = L(Hpq) = LHpq, and upon cancelling the common factors pq from abpq = LHpq, we get ab = LH, the desired result.
Applying this to your instance:
ab = LH
2160 = 12L
2160/12 = L = 180
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