SAT Integers Practice Questions with Answers

Mastering SAT Integers is fundamental to scoring well on the Math section, as these concepts serve as the building blocks for more complex topics like SAT Algebra Word problems. Integers are whole numbers that can be positive, negative, or zero, and the SAT frequently tests your ability to manipulate them through properties of parity, divisibility, and absolute value. Whether you are dealing with consecutive integers or prime factorization, understanding these core principles will help you navigate both the calculator and no-calculator portions of the exam with confidence.

Concept Explanation

SAT Integers are the set of all whole numbers, including positive numbers, negative numbers, and zero, but excluding any fractions or decimals. This set is denoted as $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$. When preparing for the College Board SAT, you must be comfortable with several specific properties of integers:

• Parity (Even and Odd): Even numbers are divisible by 2 (e.g., $-4, 0, 2, 10$), while odd numbers leave a remainder of 1 when divided by 2 (e.g., $-3, 1, 5, 11$). Key rules include:
  • Even $\pm$ Even = Even
  • Odd $\pm$ Odd = Even
  • Even $\pm$ Odd = Odd
  • Even $\times$ Any Integer = Even
  • Odd $\times$ Odd = Odd
• Consecutive Integers: These are integers that follow one another in order, such as $n, n+1, n+2$. If the problem specifies consecutive even or odd integers, the sequence is $n, n+2, n+4$.
• Divisibility and Factors: An integer $x$ is a factor of $y$ if $y$ can be divided by $x$ without a remainder. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12.
• Prime Numbers: A prime number is a positive integer greater than 1 that has exactly two factors: 1 and itself. Note that 2 is the only even prime number, and 1 is not prime.

Understanding these definitions is essential before moving into SAT Word Problems where these constraints are often hidden in the text. For a deeper look at number sets, Wikipedia's entry on Integers provides a rigorous mathematical background.

Solved Examples

Review these fully worked examples to see how SAT Integers theory is applied in practice.

1. Example 1: Parity Logic
   If $x$ is an even integer and $y$ is an odd integer, which of the following must be an odd integer?
   A) $x + 2y$
   B) $2x + y$
   C) $xy$
   D) $x^y$
   Solution:
   1. Analyze option A: Even + 2(Odd) = Even + Even = Even.
   2. Analyze option B: 2(Even) + Odd = Even + Odd = Odd. This matches our requirement.
   3. Analyze option C: Even $\times$ Odd = Even.
   4. Analyze option D: Even raised to any positive power is Even.
   5. Conclusion: The correct answer is B.
2. Example 2: Consecutive Integers
   The sum of three consecutive integers is 72. What is the largest of these integers?
   Solution:
   1. Let the three integers be $n$, $n+1$, and $n+2$.
   2. Set up the equation: $$n + (n+1) + (n+2) = 72$$
   3. Simplify: $$3n + 3 = 72$$
   4. Subtract 3: $$3n = 69$$
   5. Divide by 3: $$n = 23$$
   6. The largest integer is $n+2 = 23 + 2 = 25$.
3. Example 3: Prime Factorization
   How many distinct prime factors does the number 60 have?
   Solution:
   1. Break 60 into factors: $60 = 6 \times 10$.
   2. Break those into primes: $6 = 2 \times 3$ and $10 = 2 \times 5$.
   3. Combine: $60 = 2^2 \times 3 \times 5$.
   4. Identify distinct primes: The primes are 2, 3, and 5.
   5. Count: There are 3 distinct prime factors.

Practice Questions

1. If $k$ is an odd integer, which of the following must be an even integer?
   A) $k^2 + k$
   B) $k^2 + 2$
   C) $2k + 1$
   D) $3k - 2$
2. The sum of five consecutive odd integers is 135. What is the smallest of these integers?
3. If $x$ and $y$ are positive integers such that $3x + 2y = 18$, what is one possible value of $x$?

4. What is the least common multiple (LCM) of 12, 15, and 20?
5. If $n$ is a negative integer, which of the following has the greatest value?
   A) $n + 1$
   B) $n - 1$
   C) $-2n$
   D) $n^2$
6. A set of four consecutive even integers has a sum of 44. What is the second smallest integer in the set?
7. If $p$ is a prime number and $13 < p < 20$, what is the value of $p$?
8. How many integers between 10 and 50 are divisible by both 3 and 4?
9. If $a$ and $b$ are integers such that $a^2 = 64$ and $b^3 = 27$, what is the maximum possible value of $a - b$?
10. The product of two consecutive positive integers is 156. What is the sum of these two integers?

Answers & Explanations

1. Answer: A. If $k$ is odd, then $k^2$ is also odd (Odd $\times$ Odd = Odd). Adding an odd number to an odd number ($k^2 + k$) always results in an even number. For example, if $k = 3$, $3^2 + 3 = 12$.
2. Answer: 23. Let the integers be $n, n+2, n+4, n+6, n+8$. The sum is $5n + 20 = 135$. Subtracting 20 gives $5n = 115$. Dividing by 5 gives $n = 23$.
3. Answer: 2 or 4. If $x=2$, $3(2) + 2y = 18 \rightarrow 6 + 2y = 18 \rightarrow 2y = 12 \rightarrow y = 6$. If $x=4$, $3(4) + 2y = 18 \rightarrow 12 + 2y = 18 \rightarrow 2y = 6 \rightarrow y = 3$. Both 2 and 4 are valid integer solutions for $x$. This is similar to constraints found in SAT Linear Equations.
4. Answer: 60. Prime factors: $12 = 2^2 \times 3$, $15 = 3 \times 5$, $20 = 2^2 \times 5$. The LCM takes the highest power of each prime: $2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60$.
5. Answer: D. Since $n$ is a negative integer (e.g., $-2$), $n+1 = -1$, $n-1 = -3$, $-2n = 4$, and $n^2 = 4$. However, for larger negative integers like $-5$, $n^2 = 25$ while $-2n = 10$. $n^2$ grows much faster and will be the greatest.
6. Answer: 10. Let the integers be $n, n+2, n+4, n+6$. Sum: $4n + 12 = 44$. $4n = 32$, so $n = 8$. The integers are 8, 10, 12, 14. The second smallest is 10.
7. Answer: 17 or 19. Prime numbers between 13 and 20 are 17 and 19. Either value satisfies the condition.
8. Answer: 3. An integer divisible by both 3 and 4 must be divisible by their LCM, which is 12. The multiples of 12 between 10 and 50 are 12, 24, 36, and 48. There are 4 such integers. (Correction: Question asks for "between 10 and 50", list is 12, 24, 36, 48. Total count is 4).
9. Answer: 5. If $a^2 = 64$, then $a = 8$ or $a = -8$. If $b^3 = 27$, then $b = 3$. To maximize $a - b$, we use the largest $a$: $8 - 3 = 5$.
10. Answer: 25. Let the integers be $n$ and $n+1$. $n(n+1) = 156$. Solving $n^2 + n - 156 = 0$ using the quadratic formula or factoring gives $(n+13)(n-12) = 0$. Since they are positive, $n = 12$. The integers are 12 and 13. Sum: $12 + 13 = 25$. You can check similar logic in SAT Quadratic Equations.

Quick Quiz

Frequently Asked Questions

Is zero an even or odd integer?

Zero is an even integer because it can be divided by 2 without leaving a remainder ($0 \div 2 = 0$). It follows all algebraic rules for even numbers, such as even plus even equals even.

What is the difference between an integer and a whole number?

Integers include all whole numbers ($0, 1, 2, ...$) as well as their negative counterparts ($-1, -2, -3, ...$). While all whole numbers are integers, not all integers (specifically negative ones) are whole numbers.

Are prime numbers always odd?

No, not all prime numbers are odd; the number 2 is the only even prime number. All other prime numbers are odd because any other even number would be divisible by 2.

How do you find the sum of consecutive integers quickly?

You can find the sum of consecutive integers by multiplying the average of the first and last terms by the total number of terms. For example, the sum of 1 through 10 is $\frac{1+10}{2} \times 10 = 55$.

Can an integer be a fraction?

By definition, an integer cannot be a fraction unless that fraction simplifies to a whole number, such as $\frac{8}{2} = 4$. Numbers like $\frac{1}{2}$ or $0.75$ are not integers.

What does "distinct prime factors" mean?

Distinct prime factors are the unique prime numbers that multiply together to form a given number, regardless of how many times they repeat. For example, the distinct prime factors of 12 ($2 \times 2 \times 3$) are just 2 and 3.

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