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(male narrator) In this video,
we will take a look at solving consecutive integer problems
where the integers are either even or odd.
As we consider even numbers, we must remember
that the space between even numbers is 2.
For example, 8, 10, and 12 are even numbers with a space of 2.
Again, we will call the first number x.
To get the second number to represent that space of 2,
we will add 2 to the second number.
To get the third, we will add 2 more--or a total of 4.
Interestingly, when we look at odd numbers,
consecutive odd numbers also have a space of 2 between them.
For example, 7, 9, and 11 also have that same space of 2.
So similarly, we will call the first consecutive integer x.
To represent the next with a space of 2, we add 2.
The third, we add 2 more, or 4.
In this way, consecutive even and odd integers
are set up and solved the same way.
The only difference is the final answer.
Let's try this problem here.
Here, we are asked to find three consecutive even integers.
We will clearly label them as first, second, and third.
The first, we always represent with x.
Because they are even, we know the space is 2,
so we add 2 and 2 more to get the second and third.
We want the sum to be 84, so we will add them together
by combining like terms.
We have 3x plus 6, equals the sum--or 84.
Now we have an equation, we can quickly solve
by subtracting 6 from both sides: 3x equals 78.
Finally, to get the x alone, we divide both sides by 3,
and x is equal to 26-- an even number.
Again, this is our first number: 26.
The second number is two more than that: 28.
The third number is four more than that: 26 plus 4 is 30.
Notice, we have three consecutive even integers,
whose sum is 84.
Let's try one with odd integers.
In this problem,
we are asked to find four consecutive odd integers.
We will clearly label the first, second, third, and fourth
using our variables.
Again, we don't know the first, and to do the second,
we add the space between odd numbers.
The space is 2.
To get the third, we add 2 more, or 4.
To find the fourth, we add 2 more, or 6.
Now that they're labeled, we know their sum is 152.
We combine like terms to find 4x plus 12, equals 152.
Subtracting 12 from both sides of the equation
will give us 4x equals 140.
Finally, to get the x alone, we divide both sides by 4.
We find x is equal to 35...
an odd number, just like we expect.
The first number is x: 35.
The second number, we add 2 to x: 35 plus 2 is 37.
To find the third number, we add 4 to x: 35 plus 4 is 39.
To find the fourth number, we add 6: 35 plus 6 is 41.
Notice, we have four consecutive odd integers,
whose sum is 152.
By clearly labeling odd and even integers
with the space of 2, 4, and 6 for every problem,
we can quickly find an equation
we can solve to find our numbers.