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📝 Consecutive even integers (15 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 15 questions available

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What is Consecutive even integers?

Definition:
Consecutive even integers are even numbers that follow each other, such as 2, 4, 6, or -4, -2, 0. They differ by 2. If the first even integer is $x$ , the next are $x+2$ , $x+4$ , $x+6$ , etc. Problems often involve their sums or products.

Working:
For example, 'The sum of two consecutive even integers is 54.' Let the first be $x$ , the second $x+2$ . Equation: $x + (x+2) = 54$ . Simplify: $2x+2=54$ , subtract 2: $2x=52$ , divide: $x=26$ . The integers are 26 and 28.

Example:
Problem: 'Find three consecutive even integers whose sum is 72.' Let $x, x+2, x+4$ . Equation: $x + (x+2) + (x+4) = 72$ , $3x+6=72$ , $3x=66$ , $x=22$ . Integers: 22, 24, 26.

Reason:
This helps students handle patterns involving even numbers and reinforces the concept of using variables to represent structured sequences.

Easy

Medium

Hard

📝 All Consecutive even integers MCQs

Q1. Two consecutive even integers have a sum of 74. Which pair satisfies the condition?

A.34 and 40

B.36 and 38 ✅

C.32 and 42

D.35 and 39

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Let the first even integer be $x$ . The next consecutive even integer is $x+2$ , not $x+1$ . Thus $x+(x+2)=74$ , giving $2x=72$ and $x=36$ . Therefore, the integers are 36 and 38.

Q2. Which algebraic expression correctly represents the product of two consecutive even integers if the smaller integer is $x$ ?

x(x+1)

x(x+2)

✅

2x(x+1)

x^2+2

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Even integers occurring consecutively differ by 2. If $x$ is the smaller even integer, the next one must be $x+2$ . Their product is therefore $x(x+2)$ . The other expressions either use the wrong difference or do not represent a product.

Q3. A teacher says, "If two consecutive even integers have a sum of 126, divide 126 by 2 to get the smaller integer." What is the best evaluation of this reasoning?

A.It is always correct for consecutive even integers

B.It gives the larger integer

C.It gives the average, not the smaller integer ✅

D.It fails because even integers cannot have an odd difference

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: For two consecutive even integers $x$ and $x+2$ , their sum is $2x+2$ . Half the sum is $x+1$ , which lies exactly between the two integers. Therefore, dividing by 2 gives their average, not either integer.

Q4. Three consecutive even integers have a sum of 114. A student models them as $x,x+2,x+4$ . Which conclusion follows from the model?

A.The integers are 36, 38, and 40 ✅

B.The integers are 38, 40, and 42

C.The integers are 34, 38, and 42

D.The integers are 35, 37, and 42

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The model correctly preserves the difference of 2 between consecutive even integers. Substituting gives $x+(x+2)+(x+4)=114$ , so $3x+6=114$ , $x=36$ . Hence the three integers are 36, 38, and 40.

Q5. The sum of two consecutive even integers is 58. Which statement best explains why their average is an odd integer?

A.Every even integer has an odd average

B.The two integers differ by 2, so their midpoint is one integer above the smaller ✅

C.The sum of two even integers must be odd

D.The average is always equal to the larger integer

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: If the integers are $x$ and $x+2$ , their average is $(x+x+2)/2=x+1$ . Since $x$ is even, $x+1$ is odd. Thus the midpoint between consecutive even integers is always an odd integer.

Q6. A rectangular garden has an area of 288 square meters. Its length and width are consecutive even integers, with the length larger. What are its dimensions?

A.14 m by 20 m

B.16 m by 18 m ✅

C.12 m by 24 m

D.10 m by 28 m

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Let the width be $x$ , so the length is $x+2$ . The area condition gives $x(x+2)=288$ , or $x^2+2x-288=0$ . Factoring gives $(x-16)(x+18)=0$ . The positive dimensions are 16 m and 18 m.

Q7. A coach records the scores of two consecutive even rounds as $x$ and $x+2$ . Their total score is 86. If a bonus of 5 points is added to the smaller score, what is the new total?

A.89

B.91 ✅

C.93

D.96

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The original scores satisfy $x+(x+2)=86$ , so $2x+2=86$ and $x=42$ . The scores are 42 and 44. Adding 5 to the smaller score changes the total from 86 to $42+5+44=91$ .

Q8. A student solves $x+(x+2)=92$ as $2x+2=92$ , then writes $x=46$ . What is the error?

A.The student should subtract 2 before dividing by 2 ✅

B.The student should add 2 before dividing

C.The next even integer should be

x+1

D.The equation should contain three integers

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: From $2x+2=92$ , subtracting 2 first gives $2x=90$ , and then dividing by 2 gives $x=45$ . Although the equation correctly models consecutive even integers, $x=45$ is odd, showing that 92 cannot be the sum of two consecutive even integers.

Q9. A graph represents $y=x+(x+2)$ for integer values of $x$ . The horizontal axis shows $x$ , and the vertical axis shows the sum. Which point corresponds to two consecutive even integers whose sum is 50?

(22, 50)

(24, 50)

✅

(25, 50)

(26, 50)

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The graph represents $y=2x+2$ . Setting $y=50$ gives $2x+2=50$ , so $2x=48$ and $x=24$ . The pair is 24 and 26. Therefore, the corresponding point is $(24,50)$ , making option B correct.

Q10. A graph of $y=x+(x+2)$ is a straight line. A student claims that the point where $y=70$ gives $x=35$ , so the integers are 35 and 37. How should the graph be interpreted?

A.The claim is correct because 35 is the midpoint

B.The point gives

x=34

, producing the even pair 34 and 36 ✅

C.The point gives

x=36

, producing the pair 36 and 38

D.No point with

y=70

can exist

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The graph follows $y=2x+2$ . For $y=70$ , $2x+2=70$ , so $x=34$ . The two consecutive even integers are therefore 34 and 36. The student's value 35 is the average of the pair, not the smaller integer.

Q11. A number puzzle says that the sum of two consecutive even integers is 102 and their difference is 2. Which method is most efficient for finding the integers?

A.Use the sum to find their average, then move 1 in each direction ✅

B.Divide the sum by 2 and use the result as the smaller integer

C.Subtract 2 from the sum and divide by 2

D.Multiply the sum by 2, then subtract 2

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The average of two consecutive even integers is halfway between them and therefore odd. Since the sum is 102, the average is 51. Moving 1 in each direction gives 50 and 52, which differ by 2 and sum to 102.

Q12. A farmer has three consecutive even numbers of trees planted in three rows. The middle row has 6 more trees than the first, and the total is 150. Which set is possible?

A.46, 50, 54

B.48, 50, 52 ✅

C.46, 52, 52

D.44, 50, 56

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Three consecutive even integers must increase by exactly 2 each time. Let them be $x,x+2,x+4$ . Their sum gives $3x+6=150$ , so $x=48$ . Thus the rows contain 48, 50, and 52 trees, satisfying both the total and consecutive-even conditions.

Q13. Two consecutive even integers have a product of 168. A student claims the numbers must be 12 and 14. Which verification best supports the student's answer?

A.Their difference is 2 and their product is 168 ✅

B.Their sum is 26, which is even

C.Both numbers are divisible by 2, so they are consecutive

D.Their average is 14, so 12 and 14 are consecutive

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The defining conditions require both numbers to be even, consecutive, and to have the stated product. The pair 12 and 14 differs by 2 and has product $12\times14=168$ . The other choices do not fully verify the product and consecutiveness.

Q14. A puzzle asks for two consecutive positive even integers whose product is 360 and whose sum is 38. Which conclusion is correct?

A.18 and 20 satisfy both conditions ✅

B.16 and 22 satisfy both conditions

C.No such consecutive even integers exist

D.10 and 36 satisfy both conditions

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The pair 18 and 20 consists of consecutive even integers because their difference is 2. Their product is $18\times20=360$ , and their sum is $18+20=38$ . Thus both conditions are satisfied exactly.

Q15. Find three positive consecutive even integers whose sum equals their product divided by 20. Which set satisfies this unusual condition?

A.38752 ✅

B.39544

C.40337

D.41131

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: For 2, 4, and 6, the sum is $12$ , while the product is $48$ . Dividing the product by 20 gives $2.4$ , so this set does not satisfy the condition. Testing the structured possibilities shows that none of the listed sets works, revealing that the correct conclusion is that no option satisfies the stated relationship.

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