# If the sum of the three numbers is 27 and their product is 720. What are their numbers?

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• 720=(2^2)(3^2)10

8+9+10=27

=>

the 3 numbers are

8,9,10.

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• Both 720 and 27 are divisible by 9.  27 - 9 = 18, so the other 2 numbers are 8 and 10 OR 720 / 9 = 80.  80 = 10 X 8 Now it's a case of adding 8 + 9+ 10

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• There is not enough information. You could use trial and error and maybe some logic

eg

Divide 720 by 10 = 72

What makes 72?..... 8 x 9

ANS 8, 9 and 10.

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• a + b + c = 27

a*b*c = 720Not enough information...There are multiple solutionsAssuming you mean all three are whole numbersand all three are different.you know that each is less than 27and that720 = 2*2*2*2*3*3*5Test all the possible combinations:You will eventually reach 8, 9, 10

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• With deference to Puzzling's and nbsale's thorough answers, here is how my math-competition thinking went.

I know that to maximize the product of any n positive numbers that add up to a fixed sum, the numbers must be equal.  It's like if you have a fixed perimeter for a rectangle, the max area is a square.  720 is close to 1000, which is 10*10*10, and 10+10+10 is 30, which is darn close to 27.  So my numbers better be close to equal.  In fact, 9^3 = 729, which is already close.  Since I need a 0 on the end of 720, I'll try 10 as one of the numbers.  That leaves 72 as the product of the other two numbers, which still need to be close to 9.  How about 9 x 8?  Lucky, that works.

That doesn't get me all solutions by a long shot, but got the low-hanging fruit of a positive integer solution.  Only useful for a speed event.  Ding!  Hit the button!

• I did a similar thing to find 8, 9, 10 -- thinking 27/3 = 9 and 9^3 = 729, so figured that 9 was the middle factor. That leaves a product of two integers (one higher and one lower) that equal 80. And it's pretty obvious that we have 8 and 10.

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• The three numbers are 8, 9, and 10.

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• By trial and error  8 , 9, 10

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• There are infinitely many answers over the real numbers.

Even with integers, there isn't a unique answer.

Here's one set of integers that work:

-4, -5, 36

Sum: -4 + -5 + 36 = 27

Product: -4 * -5 * 36 = 720

And here's one with positive integers:

8, 9, 10

Sum: 8 + 9 + 10 = 27

Product: 8 * 9 * 10 = 720

• Yes, these are the only integer solutions. And the last one is the only *positive* integer solution.

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• The numbers are 37, -2.646, -7.354

• Close, but not really.  The product of those is 719.971308, and this is the math section, not engineering.

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• there are many solutions

8, 9 and 10 is the obvious one

you didn't say "integers"

xyz = 720

x+y+z = 27

try x = 7 (no solution)

try x = 8.5

yz = 84.70588

y+z = 18.5

y = 18.5–z

(18.5–z)z = 84.70588

z² – 18.5z + 84.70588 = 0

z = 10.175538, 8.32446

y = 18.5–z

y = 8.324462, 10.17554

there you are, 2 more solutions...

check

8.5•10.175538•8.324462 = 720

8.5+10.175538+8.324462 = 27

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