Find the lcm of the following number by finding their hcf first 112 and 252
Answers
Answer:
As Vinod pointed out, the product of HCF and LCM equals the product of the two given numbers, which numbers we will call x and y. As all others have pointed out so far, there is more than one way — unless the two numbers are known to be coprime (no factor in common), as correctly pointed out by Ashutosh.
Let’s say the resulting product has canonical factorization (so, looks like):
p1a1∗p2a2∗…∗pnan
What must x look like?
Well, it must look similar to this but with the powers on p1, p2, … , pn each being a choice. Specifically, let’s look at x. We find x’s possible factor of p1 has p1^e1 where 0<=e1<=a1 . In other words, as long as the power of p1 is between 0 and original a1, that power is possible. There are (a1 + 1) choices. Note that the choice of e1=0 basically means that p1 is not a factor of x.
Now, the total number of possibilities for x is found by multiplying the number of possibilities for the power of p1, the number for p2, etc.
Hence, the total number of possibilities is just
(a1+1)*(a2+1)*…*(an+1).
(In number theory, this is just the standard formula for the tau function.)
Now, this would be correct except that the pairing x ←->y could have symmetrical results. For instance, if we had x = 8 and y = 12 as possible, then the above would separately count x = 12 and y = 8. I don’t think we consider order important so (8, 12) is viewed as essentially being same as (12, 8) here.
So, every pair (x, y) that is a solution would have (y, x) as another solution. So, all we need do is divide by 2, right? Well, almost… If we are doing the pairing, there is a possibility of x = y, in which case, (y, x) would NOT be different. Now, this occurs if and only if x*y = LCM*HCF is a perfect square.
A good way to remedy this would be to take half, as before, but then round up if needed. Our conclusion is to take half of the product from above, and if this is not whole, round up. If whole, then no need to do anything else.
Just to illustrate, let’s say HCF = 12 and LCM = 18. Their product is 12*18 =216. We factor 216 as (2^3)*(3^3). There are (3+1)*(3+1) = 16 possibilities. Now, 216 is NOT a perfect square, so dividing by 2 is all we need to fix our answer. Hence, 8. As a check, here are the possibilities for (x, y) but with x<=y to avoid duplicates:
(1, 216), (2, 108), (3, 72), (4, 54), (6, 36), (8, 27), (9, 24), (12, 18).
Explanation:
L.C.M. 7056 ans. and H.C.F. 28 ans.