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