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Sums and Products (Posted on 2003-03-12) Difficulty: 2 of 5
The product of three consecutive numbers when divided by each of them in turn gives three quotients. The sum of these three quotients is equal to 74.

What are the numbers ?

See The Solution Submitted by Ravi Raja    
Rating: 2.5556 (9 votes)

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Hints/Tips Another Solution | Comment 2 of 11 |
(In reply to Solution by fwaff)

If you realize that 5*5 + 5*5 + 5*5 is 75 and the only thing that will change when using 4 and 6 is subtract 1 (see below), you can conclude that way as well.



First of all, if 2 numbers are each d from x, the product of the numbers is (x*x)-(d*d). So (5*5)-(1*1) is 1 less than (5*5)

(x+d)(x-d)=(x*x)-(xd)+(xd)-(d*d)=(x*x)-(d*d)


And (5*5)+(5*5)= ((5+5)*5) = ((4+6)+5) = (4*5) + (6*5).

Adding the two parts together: (5*5)+(5*5)+(5*5) = (4*5) + (6*5) + (4*6) + 1

This might seem more complicated, but it also shows that the product of 3 consecutive numbers is 1 less than the 3 times the middle number squared.
  Posted by Gamer on 2003-03-12 10:09:59

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