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📝 Consecutive Integers in algebra (12 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 12 questions available

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What is Consecutive Integers in algebra?

Definition:
Consecutive integers in algebra are numbers that follow each other in order without gaps, differing by 1. If the first integer is $x$ , then the next consecutive integers are $x+1$ , $x+2$ , $x+3$ , and so on. Problems often involve their sum or product.

Working:
For example, 'The sum of three consecutive integers is 72.' Let the first be $x$ , so integers are $x, x+1, x+2$ . Equation: $x + (x+1) + (x+2) = 72$ . Simplify: $3x + 3 = 72$ , subtract 3: $3x = 69$ , divide: $x = 23$ . The integers are 23, 24, 25.

Example:
Problem: 'Find three consecutive integers whose sum is 60.' Let $x$ be first, equation: $x + (x+1) + (x+2) = 60$ , $3x+3=60$ , $3x=57$ , $x=19$ . Integers: 19, 20, 21.

Reason:
This is a common type of number problem that tests understanding of sequential relationships and equation formulation.

Easy

Medium

Hard

📝 All Consecutive Integers in algebra MCQs

Q1. Three consecutive integers have a sum of $72$ . Which equation correctly models the situation if the smallest integer is $x$ ?

x+x+1+x+2=72

✅

x+x+2+x+3=72

3x+1=72

x+(x+1)+(x+3)=72

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: If the smallest integer is $x$ , the next two consecutive integers must be $x+1$ and $x+2$ . Adding them gives $x+(x+1)+(x+2)=72$ , so option A correctly represents both the consecutive relationship and the total.

Q2. Which statement best distinguishes three consecutive integers from three integers that merely differ by some amount?

A.Their average must always be zero.

B.Each integer differs from the previous integer by exactly

1

. ✅

C.Their product must always be positive.

D.The largest integer must be twice the smallest.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Consecutive integers are defined by a constant difference of exactly $1$ . Therefore, if the first integer is $x$ , the next two are $x+1$ and $x+2$ . The other statements are not required properties.

Q3. A student models three consecutive integers as $x-1$ , $x$ , and $x+1$ . Another student uses $x$ , $x+1$ , and $x+2$ . Which conclusion is correct?

A.Only the second model can represent consecutive integers.

B.Only the first model can represent consecutive integers.

C.Both models represent consecutive integers, but they use different choices for the variable. ✅

D.Neither model represents consecutive integers because the expressions contain variables.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Both models correctly describe three consecutive integers. The first chooses $x$ as the middle integer, while the second chooses $x$ as the smallest. A valid algebraic model can use different reference integers as long as the relationships remain correct.

Q4. The sum of four consecutive integers is $154$ . Without solving each integer separately first, which expression can be used to find the smallest integer $x$ ?

4x+6=154

✅

4x+4=154

x+(x+1)=154

4x+3=154

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: If $x$ is the smallest integer, the four numbers are $x$ , $x+1$ , $x+2$ , and $x+3$ . Their sum is $4x+6$ , so the correct model is $4x+6=154$ .

Q5. A theater sells tickets numbered with three consecutive seat numbers. Their sum is $189$ . Which set of seat numbers could satisfy the condition?

61,63,65

62,63,64

✅

60,63,66

59,64,66

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For three consecutive integers with sum $189$ , their average must be $189\div3=63$ . The numbers immediately surrounding $63$ are $62$ and $64$ , giving $62+63+64=189$ . Thus option B satisfies both conditions.

Q6. A student claims that if three consecutive integers have a sum divisible by $3$ , then the middle integer must be divisible by $3$ . Which example most effectively tests this claim?

4,5,6

✅

7,8,9

10,11,12

13,14,15

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The numbers $4,5,6$ are consecutive and their sum is $15$ , which is divisible by $3$ , but the middle integer $5$ is not divisible by $3$ . This counterexample disproves the student's claim.

Q7. A rectangular display uses three consecutive integer widths in three adjacent sections: $x$ , $x+1$ , and $x+2$ meters. If the total width must be $45$ meters, what is the width of the largest section?

A.14 m

B.15 m

C.16 m ✅

D.17 m

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The total width is modeled by $x+(x+1)+(x+2)=45$ , giving $3x+3=45$ and $x=14$ . Therefore, the three widths are $14,15,16$ , making the largest section $16$ meters.

Q8. A student solves $x+(x+1)+(x+2)=96$ and obtains $x=31$ . Which statement best evaluates the solution?

A.It is correct because

31+32+33=96

. ✅

B.It is incorrect because consecutive integers must begin with an even number.

C.It is incorrect because the equation should contain

x+3

D.It is correct only if

x

represents the largest integer.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Substituting $x=31$ gives $31+32+33=96$ , so the student's solution is correct. The three values differ by exactly $1$ , and $31$ is appropriately interpreted as the smallest integer in the model.

Q9. A learner writes $x+(x+1)+(x+2)=57$ , but then calculates $3x+2=57$ . What is the error?

A.The learner multiplied

x+1

2

B.The learner should have used

3x+3

, because the constants are

0,1,

and

2

. ✅

C.The learner should have changed

57

58

D.The learner incorrectly assumed the integers were negative.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Combining like terms gives $x+x+x=3x$ , while the constants add to $0+1+2=3$ . Therefore the correct simplified equation is $3x+3=57$ . The learner omitted one unit when combining the constants.

Q10. A puzzle gives three consecutive integers whose sum is $102$ . One method sets the smallest integer equal to $x$ ; another sets the middle integer equal to $m$ . Which comparison is correct?

A.Both methods must produce different integer sets.

B.The first method gives

x=34

, while the second gives

m=34

, producing the same set. ✅

C.The first method gives

x=33

, while the second gives

m=35

D.Only the method using the smallest integer can solve the problem.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Using the smallest integer gives $x+(x+1)+(x+2)=102$ , so $x=33$ , producing $33,34,35$ . Using the middle integer gives $(m-1)+m+(m+1)=102$ , so $m=34$ . Both methods identify the same set.

Q11. A coach records the scores of three players as consecutive integers. The product of the smallest and largest scores is $168$ , and their difference is $2$ . What is the middle score?

A.11

B.12

C.13 ✅

D.14

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Let the three consecutive scores be $x,x+1,x+2$ . The smallest and largest satisfy $x(x+2)=168$ , so $x^2+2x-168=0$ . Factoring gives $(x-12)(x+14)=0$ . The positive scores are $12,13,14$ , so the middle score is $13$ .

Q12. A student says, “The sum of any three consecutive integers is always odd because one number is odd and two are even.” Which response best identifies the flaw?

A.The statement is correct because three numbers always contain two even numbers.

B.The statement is false because consecutive integers can contain two odd numbers and one even number. ✅

C.The statement is false because consecutive integers always contain three odd numbers.

D.The statement is correct only when the smallest integer is positive.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Three consecutive integers can have either one odd and two even numbers or two odd and one even number. For example, $2,3,4$ sum to $9$ , while $3,4,5$ sum to $12$ . Therefore their sum is not always odd.

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