Question

Out of three numbers, the product of the first two is 28, that of the second and third is 77, and that of the third and first is 44, find the numbers.

Answer 1

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Answer 2

Suppose the sum of two consecutive numbers is 13. What are the numbers? To solve the problem, let the first number be x and the second number be x + 1.

Then:

→ $x + ( x + 1) = 132x + 1 = 132x = 12$

→  $x = 6$

So, your numbers are 6 and 7.

An Alternate Calculation

Suppose you had chosen your consecutive numbers differently from the start. In that case, let the first number be x - 3, and the second number be x - 4. These numbers are still consecutive numbers: one comes directly after the other, as follows:

→  $(x - 3) + (x - 4) = 132x - 7 = 132x = 20$

→ $x = 10$

Here you find that x equals 10, while in the previous problem, x was equal to 6. To clear up this seeming discrepancy, substitute 10 for x, as follows:

→ $10 - 3 = 7$

→  $10 - 4 = 6$

You then have the same answer as in the previous problem.

Sometimes it may be easier if you choose different variables for your consecutive numbers. For example, if you had a problem involving the product of five consecutive numbers, you could calculate it using either of the following two methods:

→  $x (x + 1) (x + 2) (x + 3) (x + 4)$

or

→ $(x - 2) (x - 1) (x) (x + 1) (x + 2)$

The second equation is easier to calculate, however, because it can take advantage of the properties of the difference of squares.

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Try these consecutive number problems. Even if you can figure out some of them without the methods discussed previously, try them using consecutive variables for practice:

• Four consecutive even numbers have a sum of 92. What are the numbers?
• Five consecutive numbers have a sum of zero. What are the numbers?
• Two consecutive odd numbers have a product of 35. What are the numbers?
• Three consecutive multiples of five have a sum of 75. What are the numbers?
• The product of two consecutive numbers is 12. What are the numbers?
• If the sum of four consecutive integers is 46, what are the numbers?
• The sum of five consecutive even integers is 50. What are the numbers?
• If you subtract the sum of two consecutive numbers from the product of the same two numbers, the answer is 5. What are the numbers?
• Do there exist two consecutive odd numbers with a product of 52?
• Do there exist seven consecutive integers with a sum of 130?

Solutions

• 20, 22, 24, 26

• -2, -1, 0, 1, 2

• 5, 7

• 20, 25, 30

• 3, 4

• 10, 11, 12, 13

• 6, 8, 10, 12, 14

• -2 and -1 OR 3 and 4

No. Setting up equations and solving leads to a non-integer solution for x.

No. Setting up equations and solving leads to a non-integer solution for x.