LCM Riddle 3

Number Theory Level 3

The positive integers A A and B B have a lowest common multiple of 300. What's the smallest possible value for A + B ? A+B ?


The answer is 37.

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Jason Dyer by Jason Dyer

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1 solution

300 = 2 2 × 3 × 5 2 300=2^2 \times 3 \times 5^2

So one of A or B must be divisible by 4, 3, 25.

Also, we have to minimize the sum. This can be done if the smaller factors are taken together and the larger ones are left alone.

So the numbers must be 4 × 3 = 12 4 \times 3 = 12 and 25 giving sum 37.

4 Helpful 0 Interesting 0 Brilliant 0 Confused

To intuitively see why we must take smaller factors together, let us consider all the possible cases of the answer. Since all of them must have the same product, they have the same G.M. we need to minimize the A.M. This usually happens if both A and B are equal but since we restrict ourselves to integers, A.M will be minimized if A and B are as close to each other. And this happens when we group the smaller factors and larger ones are left alone.

Ajinkya Shivashankar - 4 years, 4 months ago

Can we generalize this ?

Aniruddha Bagchi - 4 years, 4 months ago

Your factorization of 300 300 is missing a factor of 5 5 .

Abhishek Sinha - 4 years, 4 months ago

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Oooops , Edited it.

Ajinkya Shivashankar - 4 years, 4 months ago

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