Problems related to integer numbers are presented along with their solutions.
Problem 1:
Find two consecutive integers whose sum is equal 129.
Solution to Problem 1:

Let x and x + 1 (consecutive integers differ by 1) be the two numbers. Use the fact that their sum is equal to 129 to write the equation
x + (x + 1) = 129

Solve for x to obtain
x = 64

The two numbers are
x = 64 and x + 1 = 65

We can see that the sum of the two numbers is 129.
Problem 2:
Find three consecutive integers whose sum is equal to 366.
Solution to Problem 2:

Let the three numbers be x, x + 1 and x + 2. their sum is equal to 366, hence
x + (x + 1) + (x + 2) = 366

Solve for x and find the three numbers
x = 121 , x + 1 = 122 and x + 2 = 123
Problem 3:
The sum of three consecutive even integers is equal to 84. Find the numbers.
Solution to Problem 3:

The difference between two even integers is equal to 2. let x, x + 2 and x + 4 be the three numbers. Their sum is equal to 84, hence
x + (x + 2) + (x + 4) = 84

Solve for x and find the three numbers
x = 26 , x + 2 = 28 and x + 4 = 30

The three numbers are even. Check that their sum is equal to 84.
Problem 4:
The sum on an odd integer and twice its consecutive is equal to equal to 3757. Find the number.
Solution to Problem 4:

The difference between two odd integers is equal to 2. let x be an odd integer and x + 2 be its consecutive. The sum of x and twice its consecutive is equal to 3757 gives an equation of the form
x + 2(x + 2) = 3757

Solve for x
x = 1251

Check that the sum of 1251 and 2(1251 + 2) is equal to 3757.
Problem 5:
The sum of the first and third of three consecutive odd integers is 131 less than three times the second integer. Find the three integers.
Solution to Problem 5:

Let x, x + 2 and x + 4 be three integers. The sum of the first x and third x + 4 is given by
x + (x + 4)

131 less than three times the second 3(x + 2) is given by
3(x + 2)  131

"The sum of the first and third is 131 less than three times the second" gives
x + (x + 4) = 3(x + 2)  131

Solve for x and find all three numbers
x = 129 , x + 2 = 131 , x + 4 = 133

As an exercise, check that the sum of the first and third is 131 less than three times
Problem 6:
The product of two consecutive odd integers is equal to 675. Find the two integers.

Let x, x + 2 be the two integers. Their product is equak to 144
x (x + 2) = 675

Expand to obtain a quadratic equation.
x^{ 2} + 2 x  675 = 0

Solve for x to obtain two solutions
x = 25 or x = 27
if x = 25 then x + 2 = 27
if x = 27 then x + 2 = 25

We have two solutions. The two numbers are either
25 and 27

or
27 and 25

Check that in both cases the product is equal to 675.
Problem 7:
Find four consecutive even integers so that the sum of the first two added to twice the sum of the last two is equal to 742.
Solution to Problem 7:

Let x, x + 2, x + 4 and x + 6 be the four integers. The sum of the first two
x + (x + 2)

twice the sum of the last two is written as
2 ((x + 4) + (x + 6)) = 4 x + 20

sum of the first two added to twice the sum of the last two is equal to 742 is written as
x + (x + 2) + 4 x + 20 = 742

Solve for x and find all four numbers
x = 120 , x + 2 = 122 , x + 4 = 124 , x + 6 = 126

As an exrcise, check that the sum of the first two added to twice the sum of the last two is equal to 742
Problem 8:
When the smallest of three consecutive odd integers is added to four times the largest, it produces a result 729 more than four times the middle integer. Find the numbers and check your answer.
Solution to Problem 8:

Let x, x + 2 and x + 4 be the three integers. "The smallest added to four times the largest is written as follows"
x + 4 (x + 4)

"729 more than four times the middle integer" is written as follows
729 + 4 (x + 2)

"When the smallest is added to four times the largest, it produces a result 729 more than four times the middle" is written as follows
x + 4 (x + 4) = 729 + 4 (x + 2)

Solve for x and find all three numbers
x + 4 x + 16 = 729 + 4 x + 8
x = 721
x + 2 = 723
x + 4 = 725

Check: the smallest is added to four times the largest
721 + 4 * 725 = 3621

four times the middle
4 * 723 = 2892

3621 is more than 2892 by
3621  2892 = 729

The answer to the problem is correct.
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