# Question

There are 3 mystery numbers. The products of 2 of the 3 numbers are 432, 540, 720. What are the 3 numbers?

By breaking the 3 products into their factors, DD and I were able to analyse and deduce that the 3 numbers are 18, 24 and 30. Question is how do we present this thinking process into words on paper?

Thank you, Acestarling and SAHMom!

Not sure if can use calculator, but thanks to your solutions, I modified them to solve using prime factorisation.

ab = 432 = 24 . 33

ac = 540 = 22 . 33 . 5

bc = 720 = 24 . 32 . 5

ab x ac x bc = a2b2c2 = 210 . 38 . 52

abc = 25 . 34 . 5

a = abc / bc = 25 . 34 . 5 / 24 . 32 . 5 = 2 . 32 = 18

b = abc / ac = 25 . 34 . 5 / 22 . 33 . 5 = 23 . 3 = 24

c = abc / ab = 25 . 34 . 5 / 24 . 33 = 2 . 3 . 5 = 30

Thank you both!

Great solution… I am teaching my son this now.

Credit goes to Acestarling and SAHMom. 🙂

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Let the numbers be a, b and c.

a x b    X    a x c    X   b x c = 432x540x720 (random order)

square root on both sides:

a x b x c = 12960

12960/432 = 30

12960/540 = 24

12960/720 = 18

assuming cannot use calculator. Use prime factorisation (or “not so prime” factorisation for ease of use)

432 X 540 X 720

= 8 X 54    X   54X 10  X      8x 10x 3×3

square root is 8 X 54 X 10×3 = 12960

8 X 54 X 10×3 /(8×54) = 30

8 X 54 X 10×3 /(54×10) = 24

8 X 54 X 10×3 /(8x10x3x3) = 18

ps: I posted same question to my child who got stuck with the question and zoomed immediately to prime factorisation of each product….

Oops…didn’t know I can’t have 2 accepted answers.

The Accept Answer function is intended for the thread starter to choose the best answer to his question.  However, we will enhance it so that questioners can replace his selected answer with another answer should a better answer come along later.