Question

The GCD(a, b) = 9, the LCM(a, b)=378. Find the least possible value of a + b.

Answer 1

Answer: 117

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Work Shown:

I'm assuming your teacher is applying the restriction that both 'a' and 'b' are positive integers.

L = LCM(a,b) = LCM of 'a' and 'b' = 9

G = GCD(a,b) = GCD of 'a' and 'b' = 378

L = (a*b)/G .... this shows how the LCM and GCD are connected

G*L = a*b

a*b = G*L

a*b = 9*378

ab = 3402

b = 3402/a

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a+b = a + (3402/a)

a+b = (a^2)/a + (3402/a)

a+b = (a^2+3402)/a

Replace 'a+b' with y. Replace every copy of 'a' with x. We get this equation

y = (x^2+3402)/x

Use a calculator or calculus to find that the lowest point occurs at the approximate location (x,y) = (58.3266662412, 116.6533325713)

Check out the diagram below. The graph was made with GeoGebra.

At this point we would stop since we found the lowest y value; however, keep in mind that 'a' and 'b' are integers. There is no way to have a+b add to the non-integer result 116.6533325713

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What we must do is look at a table of values where we see what integer x values lead to y being an integer as well. Those pairs are

x = 54, y = 117

x = 63, y = 117

Which are highlighted in the same diagram as the graph.

I restricted the table so that $50 \le x \le 60$ due to the lowest point found earlier having an x coordinate of roughly 58.3266662412

Based on what the table says, the smallest a+b can be is 117.