**Consecutive Integers**

- Find two consecutive integers whose sum is 45.

Let x = the smaller of the two numbers.

x + 1 = the larger of the two numbers.

The sum of the two numbers equals 45.

Write out the equation: x + (x + 1) = 45

2x + 1 = 45

2x = 44

x = 22

The smaller of the two numbers is 22, so the other number is 23.

Check: 22 + 23 = 45

- Find two consecutive integers whose sum is -29.

Let x = the smaller of the two numbers.

x + 1 = the larger of the two numbers.

The sum of the two numbers equals -29.

x + (x + 1) = -29

x + x + 1 = -29

2x + 1 = -29

2x = -30

x = -15

The smaller of the two numbers is -15, so the other number is -14.

Check: -15 + -14 = -29

- Find three consecutive integers whose sum is 48.

Let x = the smallest of the three numbers.

Let x + 1 = the second smallest of the three numbers.

Let x + 2 = the largest of the three numbers.

The sum of the three numbers is 48.

x + (x + 1) + (x + 2) = 48

x + x + 1 + x + 2 = 48

3x + 3 = 48

3x = 45

x = 15

The smallest of the three numbers is 15, the second smallest is 16, and the largest is 17.

Check: 15 + 16 + 17 = 48

- Find three consecutive integers whose sum is -147.

Let x = the smallest of the three numbers.

Let x + 1 = the second smallest of the three numbers.

Let x + 2 = the largest of the three numbers.

The sum of the three numbers is -147.

Write out the equation: x + (x + 1) + (x + 2) = -147

x + x + 1 + x + 2 = -147

3x + 3 = -147

3x = -150

x = -50

The smallest of the three numbers is -50, the second smallest is -49, and the largest is -48.

Check: -50 + -49 + -48 = -147

- Find two consecutive even integers whose sum is 66.

Let x = the smallest of the two numbers.

Let x + 2 = the largest of the two numbers.

The sum of the two numbers is 66.

Write out the equation: x + (x + 2) = 66

x + x + 2 = 66

2x + 2 = 66

2x = 64

x = 32

The smallest of the two numbers is 32, and the largest is 34.

Check: 32 + 34 = 66

- Find three consecutive even integers whose sum is 72.

Let x = the smallest of the three numbers.

Let x + 2 = the second smallest of the three numbers.

Let x + 4 = the largest of the three numbers.

The sum of the three numbers is 72.

x + (x + 2) + (x + 4) = 72

x + x + 2 + x + 4 = 72

3x + 6 = 72

3x = 66

x = 22

The smallest of the three numbers is 22, the second smallest is 24, and the largest is 26.

Check: 22 + 24 + 26 = 72

- Find two consecutive odd integers whose sum is 88.

Let x = the smallest of the two numbers.

Let x + 2 = the largest of the two numbers.

The sum of the two numbers is 88.

x + (x + 2) = 88

x + x + 2 = 88

2x + 2 = 88

2x = 86

x = 43

The smallest of the two numbers is 43, and the largest is 45.

Check: 43 + 45 = 88

- Find four consecutive odd integers whose sum is 56.

Let x = the smallest of the four numbers

Let x + 2 = the second smallest of the four numbers

Let x + 4 = the third smallest of the four numbers

Let x + 6 = the largest of the four numbers

The sum of the four numbers is 56.

x + (x + 2) + (x + 4) + (x + 6) = 56

x + x +2 + x + 4 + x + 6 = 56

4x + 12 = 56

4x = 44

x = 11

The smallest of the four numbers is 11, the second smallest is 13, the third smallest is 15, and the largest is 17.

Check: 11 + 13 + 15 + 17 = 56

- Find two consecutive even integers such that the sum of the larger and twice the smaller is 62

Let x = the smallest of the two numbers.

Let x + 2 = the larger of the two numbers.

(x + 2) + 2x = 62

3x + 2 = 62

3x = 60

x = 20 and is the smaller even number

22 then is the next consecutive even integer.

Check

22 + 2(20) = 62

22 + 40 = 62

62 = 62

- Find three consecutive even integers such that the sum of the smallest and the largest is 36.

Let x = the smallest of the three numbers

Let x + 2 = the second smallest of the three numbers

Let x + 4 = the largest of the three numbers

Even integers are 2 apart

x + (x+4) = 36

2x + 4 = 36

2x = 32

x = 16

So the smallest number is 16, the largest is 20 and the second largest is 18

Check

16 + 20 = 36

- Find three consecutive odd integers such that the sum of the smallest and 4 times the largest is 61

Let x = the smallest of the three numbers

Let x + 2 = the second smallest of the three numbers

Let x + 4 = the largest of the three numbers

Odd integers are 2 apart

x + 4(x+4) = 61

x + 4(x + 4) = 61

x + 4x + 16 = 61

5x + 16 = 61

5x = 45

x = 9

So the smallest number is 9, the largest is 13 and the second largest is 11

Check

9 + 4(13) = 61

9 + 52 = 61

61 = 61

- Find three consecutive integers such that the Sum of twice the smallest and 3 times the largest is 126.

Let x = the smallest of the three numbers

Let x + 1 = the second smallest of the three numbers

Let x + 2 = the largest of the three numbers

Consecutive integers are 1 apart

2x + 3(x + 2) = 126

2x + 3x + 6 = 126

5x + 6 = 126

5x = 120

x = 24

So the smallest number is 24, the largest is 26 and the second largest is 25

Check

2(24) + 3(26) = 126

48 + 78 = 126

126 = 126

- Eight more than the square of a number is the same as 6 times the number, Find the number.

Let x = the number

x^2 + 8 = 6x

x^2 -6x + 8 = 0

Factor

(x – 2) (x – 4) = 0

x = 2 or 4

Check

2^2 + 8 = 6(2)

4 + 8 = 12

12 = 12

or

4^2 + 8 = 6(4)

16 + 8 = 24

24 = 24

- Fifteen less than the square of a number is the same as twice the number. Find the number.

Let x = the number

x^2 – 15 = 2x

x^2 -2x -15 = 0

Factor

(x – 5) (x + 3) = 0

x = 5 or -3

Check

5^2 – 15 = 2(5)

25 – 15 = 10

10 = 10

or

(-3)^2 -15 = 2(-3)

9 – 15 = -6

-6 = -6

- If a number is added to twice its square, the result is 6. Find the number.

Let x = the number

2(x^2) + x = 6

2(x^2) + x -6 = 0

Factor

(2x – 3) (x + 2) = 0

x = 3/2 or -2

Check

2(3/2^2) + 3/2 = 6

2(9/4) + 3/2 = 6

9/2 + 3/2 = 6

12/2 = 6

6 = 6

or

2(-2^2) + -2 = 6

2(4) – 2 = 6

8 – 2 = 6

6 = 6

- Seven less than 4 times the square of a number is 18. Find the number.

Let x = the number

4(x^2) – 7 = 18

Solve

4(x^2) = 25

(x^2) = 25/4

Take the square root of both sides

x = 5/2 or -5/2

Check

2(3/2^2) + 3/2 = 6

2(9/4) + 3/2 = 6

9/2 + 3/2 = 6

Check

4((5/2)^2) – 7 = 18

4(25/4) – 7 = 18

25 – 7 = 18

18 = 18

or

4((-5/2)^2) – 7 = 18

4(25/4) – 7 = 18

25 – 7 = 18

18 = 18

- Find two consecutive integers whose product is 56.

Let x = the smallest of the two numbers.

Let x + 1 = the larger of the two numbers.

x (x + 1) = 56

x^2 + x = 56

x^2 + x -56 = 0

(x – 7) (x + 8) = 0

x = 7 or -8

So the two consecutive numbers are 7 and 8 or -8 and -7

Check

7(8) = 56

56 = 56

or

-7(-8) = 56

- Find two consecutive positive odd integers whose product is 35.

Let x = the smallest of the two numbers.

Let x + 2 = the largest of the two numbers.

Consecutive odd are 2 apart

x(x + 2) = 35

x^2 + 2x = 35

x^2 + 2x -35 = 0

(x – 5) (x + 7) = 0

x = 5 or -7

So the consecutive odd number are 5 and 7 or -7 and -5

We must reject -7 and -5 because the question asks for positive.

This leaves us with 5 and 7

Check

5(7) = 35

35 = 35

- The sum of the squares of two consecutive integers is 41. Find the integers.

Let x = the smallest of the two numbers.

Let x + 1 = the largest of the two numbers.

Consecutive odd are 2 apart

(x^2) + (x + 1)^2 = 41

(x^2) + (x + ) (x + 1) = 41

x^2 + (x^2 + 2x + 1) = 41

x^2 + x^2 + 2x + 1 = 41

2(x^2) + 2x + 1 = 41

2(x^2) + 2x + -40 = 0

(2x – 8) (x + 5) = 0

x = 4 or -5

So the consecutive integers are 4 and 5 or -5 and -4

Check

(4^2) + (5 + 1)^2 = 41

16 + 6^2 = 41

16 + 25 = 41

41 = 41

or

((-5)^2) + (-4 + 1)^2 = 41

25 + (-3)^2 = 41

25 + 16 =41

41 = 41

- Find two consecutive odd integers such that the square of the first, added to 3 times the second, is 24.

Let x = the smallest of the two numbers.

Let x + 2 = the largest of the two numbers.

Consecutive odd integers are 2 apart.

x^2 + 3(x + 2) = 24

x^2 + 3x + 6 = 24

x^2 + 3x – 18 = 0

(x – 3) (x + 6) = 0

x = 3 or -6

So the consecutive integers are 3 and 5 or -6 and -4

Because the problem asks for odd we must reject -6 and -4 and our answer is 3 and 5

Check

3^2 + 3(5) = 24

9 + 15 = 24

24 = 24

- Find two consecutive even integers such that the square of the second, decreased by twice the first, is 52.

Let x = the smallest of the two numbers.

Let x + 2 = the larger of the two numbers.

Consecutive even integers are 2 apart.

(x + 2)^2 – 2x = 52

x^2 + 4x + 4 – 2x = 52

x^2 + 2x + 4 = 52

x^2 + 2x – 48 = 0

Factor

(x – 6) (x + 8) = 0

x = 6 or -8

So the two consecutive even integers are 6 and 8 or -8 and -6

Check

(x + 2)^2 – 2x = 52

(8)^2 – 2(6) = 52

64 – 12 = 52

52 = 52

or

(x + 2)^2 – 2x = 52

(-6)^2 – 2(-8) = 52

36 -(-16) = 52

36 + 16 = 52

52 = 52

- Find three consecutive positive integers such that the square of the first, increased by the last, is 22.

Let x = the smallest of the three numbers

Let x + 1 = the second smallest of the three numbers

Let x + 2 = the largest of the three numbers

Integers are apart

x^2 + (x + 2) = 22

x^2 + x + 2 = 22

x^2 + x + -20 = 0

(x – 4) (x + 5) = 0

x = 4 or -5

The three consecutive integers are 4, 5 and 6 or -5, -4 and -3

We must reject -5, -4 and -3 because they are negative.

We are asked for positive integers in the problem.

Check

x^2 + (x + 2) = 22

4^2 + (6) = 22

16 + 6 = 22