Question

The sum of three consecutive integers is 24. Find the product of the integers.

Answer 1

Hint: Assume that the first integer is x and thus, write the next two integers as \[x+1\] and \[x+2\]. Write an equation relating the sum of three integers and simplify them to calculate the value of three integers. Multiply the integers to find their product.

Complete step-by-step answer:
We know that the sum of three integers is 24. We have to calculate the product of the integers.
Let us assume that the first integer is x. Thus, we have the next two integers as \[x+1\] and \[x+2\].
We know that the sum of these integers is 24. So, we have \[x+\left( x+1 \right)+\left( x+2 \right)=24\].
Simplifying the above equation, we have \[3x+3=24\].
Thus, we have \[3x=24-3=21\].
So, we have \[x=\dfrac{21}{3}=7\].
Thus, the next two integers are \[x+1=7+1=8\] and \[x+2=7+2=9\].
So, the three consecutive integers whose sum is 24 are 7, 8 and 9.
We will now calculate the product of three integers, 7, 8 and 9.
We will multiply the first two integers, i.e., 7 and 8. Then, we will multiply the resulting value with the third integer, i.e., 9.
Thus, we have \[7\times 8=56\].
So, we have \[7\times 8\times 9=56\times 9=504\].
Hence, the product of three consecutive integers whose sum is 24 is 504.

Note: We can also solve this question by taking the second integer to be ‘a’ and the first and third integer to be \[a-1\] and \[a+1\] respectively. We will write equations involving the sum of three integers and simplify them to find the three integers. We must also keep in mind that multiplication is an associative property. Thus, we can first multiply any two of the integers and then multiply the value with the third integer.