SOLUTION: If the product of four consecutive integers is equal to one of the integers, what is the largest possible value of one of the integers?

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Question 201765: If the product of four consecutive integers is equal to one of the integers, what is the largest possible value of one of the integers?
Found 2 solutions by jsmallt9, MathTherapy:
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
Let's look at the possibilities:
  1. All four integers are negative. The product of four negative integers is positive. So the product could not be equal to one of the four negative integers.
  2. A mix of positive and negative integers. These integers would not be consecutive because zero was not included.
  3. All four integers are positive. The lowest four positive integers are 1, 2, 3 and 4. The product of these is 24 which is significantly larger than any of the four integers. It should be obvious that any other combination of four consecutive positive integers would not work either because we would be replacing smaller numbers with larger ones -- 2*3*4*5 = 120; 3*4*5*6 = 360; etc. -- and get products that are farther and farther away from the four integers.
  4. The only possibility remaining is where zero is one of the four numbers. Any product involving zero will work because any product involving zero will be zero. Since the sequence of four consecutive integers must include zero, the sequence with the largest values for the other integers would have zero as the lowest of the four: 0, 1, 2, 3.
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

The only four integers whose product equals one of the four integers will include 0 as 1 of its 4 integers. Therefore, the largest integer, going to the right of 0 on the number line, will be 3, and so, the four consecutive integers are 0, 1, 2 , & 3.
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