SOLUTION: Find three consecutive positive odd integers in which the product of the smallest and the largest interger is 357

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Question 1166936: Find three consecutive positive odd integers in which the product of the smallest and the largest interger is 357
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
.

Let x be the middle odd number between the smallest and the largest.


Then you have this equation


    (x-2)*(x+2) = 357,   or

     x^2 - 4 = 357

     x^2 = 357 + 4 = 361

     x =  = 19.


ANSWER.  The three numbers are 17, 19 and 21.

Solved.

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Notice how this introduction of the middle value as a major unknown facilitates the solution.

MEMORIZE this trick --- it works and is useful in many similar problems (!)




Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Tutor @ikleyn has shown a good formal algebraic solution.

Note especially her statement that in this problem, like in many other similar problems, the formal algebraic solution is made easier by choosing your variable x to be the middle of the three numbers.

If a formal algebraic solution is not required, you can get the answer quickly, and get some good mental exercise, by solving it informally, with logical reasoning and some mental arithmetic.

Here are a couple of different ways you could do that.

(1) The three numbers are close together; that means the middle number is very close to the square root of the product of the first and last. The closest perfect square to 357 is 361, which is 19 squared. Then some checking shows that the numbers 17, 19, and 21 satisfy all the conditions of the problem.

(2) Find a factorization of 357 which is the product of two odd numbers that differ by 4:

357 = 3*119 = 3*7*17 = 21*17

The first and last of the odd integers are 17 and 21; 19 is in the middle.
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