ACT Math · Number & Quantity
Number Properties and Integers on the ACT: Every Rule Explained
Number property questions should be easy points on the ACT Math section, but they’re so unique that many students miss them. You probably haven’t seen questions like these in your math class.
Since these questions are new, students get overwhelmed. But once you learn the basic rules (and tips) in this Study Guide, Number Property questions will be easy.
There are usually 2–4 number property questions on each ACT Math section, so getting them all correct is very important.
Be sure to learn all the rules in this Study Guide, even the “Low Frequency” question types, because you do not want to miss out on free points.
Concepts covered in this guide
| Concept | Named Method / Rule | Frequency |
|---|---|---|
| Even and odd rules: addition, subtraction, and multiplication | The Parity Rules | Low |
| Zero and one as special cases every ACT student must know | The Edge Case List | Low |
| Prime numbers: definition, why 1 is not prime, and what to memorize | The Two-Divisor Rule | Very High |
| Consecutive integers: how to write them algebraically vs. consecutive even/odd | The Consecutive Setup | Medium |
| Absolute value: definition and ACT question patterns | The Distance Definition | High |
| “Must be true” questions: the plug-in strategy with critical test cases | The Four-Case Test | Low |
| Remainders: definition and the ACT remainder question format | The Remainder Formula | Low |
Concept 1
Even and Odd Rules: Addition, Subtraction, and Multiplication
Very High FrequencyEven and odd rules (called parity rules) govern whether the result of an operation between integers is even or odd. These rules apply to all integers — positive, negative, and zero — because parity is determined solely by divisibility by 2, not by sign. A negative even number is still even; a negative odd number is still odd.
Named Method
The Parity Rules
Memorize the parity rules as a reference table. When in doubt, test with small numbers: substitute 2 for even and 3 for odd.
| Operation | Even + Even | Even + Odd | Odd + Odd |
|---|---|---|---|
| Addition / Subtraction | Even | Odd | Even |
| Multiplication | Even | Even | Odd |
✓ Parity with negatives — same rules apply
✗ Common error — treating negatives as non-integers
ACT-style practice question
If m is an even integer and n is an odd integer, which of the following must be even?
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Concept 2
Zero and One as Special Cases Every ACT Student Must Know
Very High FrequencyZero and one have properties that contradict student assumptions, and the ACT exploits both. These are not obscure facts — they appear directly in answer choices and in the test cases required for “must be true” questions. Every student should have all four properties of zero and one memorized before test day.
Named Method
The Edge Case List
Zero (0):
• Zero is an integer. • Zero is even (0 = 2 × 0, divisible by 2). • Zero is neither positive nor negative — it is the boundary between them. • Zero is not prime. • Multiplying any number by zero gives zero. • Division by zero is undefined.
One (1):
• One is a positive integer. • One is odd. • One is not prime. A prime number must have exactly two distinct positive divisors: 1 and itself. The number 1 has only one divisor (itself), so it fails this requirement. • One is not composite either. It is its own category. • Multiplying or dividing any number by 1 leaves it unchanged (the identity property).
ACT-style practice question
Which of the following statements about integers is true?
Concept 3
Prime Numbers: Definition, Why 1 Is Not Prime, and What to Memorize
High FrequencyA prime number is a positive integer greater than 1 that has exactly two distinct positive divisors: 1 and itself. The number 1 is not prime because it has only one positive divisor (itself), failing the two-divisor requirement. The number 2 is the only even prime. Every other even number is divisible by 2 and therefore has at least three divisors, making it composite (not prime).
Named Method
The Two-Divisor Rule
A number is prime if and only if it has exactly two distinct positive divisors: 1 and itself. To test whether a number is prime: list all its divisors. If the list has exactly two entries (1 and the number), it is prime. If it has more, it is composite. If it has only one (which only happens with 1), it is neither.
Primes to know cold for the ACT (up to 50): 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
Common traps: 1 (not prime — only one divisor), 9 (not prime — divisible by 3), 15 (not prime — divisible by 3 and 5), 21 (not prime — divisible by 3 and 7), 27 (not prime — divisible by 3), 49 (not prime — 7×7), 51 (not prime — divisible by 3).
ACT-style practice question
Which of the following lists contains only prime numbers?
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Concept 4
Consecutive Integers: How to Write Them Algebraically vs. Consecutive Even/Odd
High FrequencyConsecutive integers increase by 1. Consecutive even integers increase by 2 (because even numbers are spaced 2 apart). Consecutive odd integers also increase by 2. The algebraic expression for each type is different, and using the wrong expression produces a setup that cannot be solved correctly regardless of the algebra that follows.
Named Method
The Consecutive Setup
Consecutive integers: n, n+1, n+2, n+3, … (increment by 1 each time).
Consecutive even integers: n, n+2, n+4, n+6, … where n is even (increment by 2 each time). Example: 4, 6, 8, 10.
Consecutive odd integers: n, n+2, n+4, n+6, … where n is odd (increment by 2 each time). Example: 3, 5, 7, 9.
Critical point: consecutive even and consecutive odd integers use the same increment (+2) but start from different values. The increment is +2 for both — not +1 for one and +2 for the other. Students who write consecutive even integers as n, n+1, n+2 have created consecutive integers, not consecutive even integers.
After solving for n, verify your answer by checking that all resulting values are actually even (for consecutive even) or all odd (for consecutive odd). If the values are not all even or all odd, the setup used the wrong increment.
✓ Consecutive even: increment by 2
✗ Common error — increment by 1 for even integers
ACT-style practice question
The sum of three consecutive even integers is 54. What is the largest of the three integers?
Concept 5
Absolute Value: Definition and ACT Question Patterns
High FrequencyThe absolute value of a number is its distance from zero on the number line, always expressed as a non-negative value. |a| = a if a ≥ 0, and |a| = −a if a < 0. When the ACT places an expression inside absolute value bars, that expression can be either positive or negative and still produce the same positive result — which means absolute value equations always have two cases to solve.
Named Method
The Distance Definition
For equations of the form |expression| = k where k > 0: split into two cases and solve both. Case 1: expression = k. Case 2: expression = −k. Both cases produce valid solutions. Never discard the negative case without checking it.
For |expression| = 0: only one case. The expression itself must equal zero.
For |expression| = k where k < 0: no solution. Absolute value is always ≥ 0, so it can never equal a negative number. If you see this in an ACT answer choice labeled “no solution,” that is correct.
ACT twist: after solving both cases, always verify each solution in the original equation. Occasionally one solution does not satisfy the original (extraneous solution), particularly when variables appear outside the absolute value bars as well.
ACT-style practice question
What are all values of x that satisfy |3x − 9| = 12?
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Concept 6
“Must Be True” Questions: The Plug-In Strategy with Critical Test Cases
Very High FrequencyACT “must be true” questions present an integer expression and ask which statement is always true regardless of which integer is substituted. To disprove a wrong answer, you only need to find one integer value that makes it false. To confirm the correct answer, the statement must hold for every integer you test. The choice of which integers to test is not random — the ACT engineers its wrong answers to be true for positive integers but false for at least one critical test case.
Named Method
The Four-Case Test
For any “must be true” question about integers, always test these four cases in this order:
Strategy
Eliminate with counterexamples
Test each answer choice against these four cases. One counterexample (a single value of n that makes the statement false) is sufficient to eliminate an answer choice. The correct answer will hold for all four cases — and for any other integer you try. If only one answer choice survives all four cases, that is the answer.
Important: if the problem says “n is an integer,” test all four cases including zero and negatives. If the problem says “n is a positive integer,” skip negative test cases but still test n = 1 (which often produces different behavior than n = 2 or higher). If the problem does not say “n is an integer,” also test n = 0.5 — non-integer values may be valid inputs.
ACT-style practice question
If n is any integer, which of the following must be true?
Concept 7
Remainders: Definition and the ACT Remainder Question Format
Medium FrequencyA remainder is the amount left over when one integer is divided by another and the division does not come out evenly. If a ÷ b = q with remainder r, then a = b × q + r, where 0 ≤ r < b. The remainder is always less than the divisor and always a non-negative integer. On the ACT, remainder questions either ask you to find a remainder directly or ask for the value of a variable given information about a remainder.
Named Method
The Remainder Formula
a = b × q + r, where a is the dividend (the number being divided), b is the divisor, q is the quotient (whole number result of division), and r is the remainder (0 ≤ r < b).
To find the remainder of a ÷ b: divide a by b using long division or a calculator. The remainder is what is left after the largest whole-number multiple of b is subtracted from a.
Calculator method: compute a ÷ b. Note the decimal portion. Multiply that decimal by b. The result is the remainder. Example: 47 ÷ 6 = 7.833… The decimal is 0.833… Multiply: 0.833… × 6 = 5. Remainder = 5.
ACT remainder patterns: “When n is divided by 7, the remainder is 3. Which of the following could be n?” Set up: n = 7q + 3 for some non-negative integer q. Test the answer choices: does (choice − 3) divide evenly by 7?
Step 1: 47 ÷ 6 = 7 remainder r (since 6 × 7 = 42)
Step 2: r = 47 − 42 = 5
Check: 47 = 6 × 7 + 5. 0 ≤ 5 < 6. ✓
ACT-style practice question
What is the remainder when 83 is divided by 9?
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Quick-Reference Summary: All 7 ACT Number Properties Concepts
| Concept | Named Method | Key Rule |
|---|---|---|
| Even and odd rules | The Parity Rules | Even × anything = even. Even ± odd = odd. Odd ± odd = even. Sign never changes parity. |
| Zero and one | The Edge Case List | Zero: even, integer, neither positive nor negative, not prime. One: odd, positive, not prime, not composite. |
| Prime numbers | The Two-Divisor Rule | Prime = exactly 2 distinct positive divisors. 1 is not prime. 2 is the only even prime. Know primes to 50. |
| Consecutive integers | The Consecutive Setup | Consecutive integers: n, n+1, n+2. Consecutive even or odd: n, n+2, n+4. Both even/odd use +2, not +1. |
| Absolute value | The Distance Definition | |expr|=k → two cases: expr=k and expr=−k. Always solve both. Verify solutions in original equation. |
| “Must be true” questions | The Four-Case Test | Test n=2, n=−2, n=0, n=1. One counterexample eliminates a choice. Correct answer survives all four. |
| Remainders | The Remainder Formula | a = b×q + r where 0 ≤ r < b. Remainder = dividend − (divisor × quotient). Always less than divisor. |
How to Approach Number Properties Questions on Test Day
Tip 1
On every “must be true” or “must be false” question, run the Four-Case Test before reading the answer choices. Test n = 2, n = −2, n = 0, and n = 1 against the condition in the question, then evaluate each answer choice against those same values. One counterexample eliminates a choice permanently. This systematic approach is faster than abstract reasoning under pressure and produces the correct answer on virtually every ACT integer property question.
Tip 2
When in doubt about parity, substitute numbers. Plug in 2 for any even variable and 3 for any odd variable, then compute the expression. The numerical result will either be even or odd, and that tells you the parity rule for the expression. This substitution approach works on every even/odd question and takes about 15 seconds — faster than reasoning from abstract rules under time pressure.
Tip 3
Before writing the algebraic expression for a consecutive integer problem, write out the first two or three actual values to confirm the increment. For consecutive even integers: write 4, 6, 8 to confirm the gap is 2, not 1. Then write n, n+2, n+4. For consecutive odd integers: write 3, 5, 7 to confirm the same. Writing two concrete examples takes five seconds and prevents the most common consecutive integer setup error — using n+1 instead of n+2.
Tip 4
On absolute value equations, always solve both cases and always verify both solutions in the original equation before reporting the answer. The ACT sometimes designs absolute value questions where one solution is extraneous (does not satisfy the original equation), and “both solutions” appears as a wrong answer choice. The verification step takes 20 seconds and is the only reliable way to catch an extraneous solution.
Common Questions About ACT Number Properties
Zero is even. The definition of an even integer is any integer that is divisible by 2 with no remainder. Zero divided by 2 equals zero with no remainder, so zero satisfies the definition. Additionally, zero fits the pattern of even integers: …, −4, −2, 0, 2, 4, … — every other integer in sequence, alternating even and odd.
Zero is also neither positive nor negative. It is an integer, but it is not a positive integer and not a negative integer. This matters on the ACT when a question says “n is a positive integer” vs. “n is an integer” — zero is only a valid test case in the second situation. Memorize the Edge Case List: zero is an even, non-positive, non-negative integer that is not prime.
Always x, x+2, x+4 for consecutive even integers — and for consecutive odd integers too. Even numbers are spaced 2 apart (2, 4, 6, 8…), so the increment between consecutive even integers is always +2. If you write x, x+1, x+2 and x is even, then x+1 is odd, which breaks the “consecutive even” condition immediately.
The easiest way to remember: write out two concrete examples before setting up the variable expression. For consecutive even integers, write 4, 6, 8. The gap is 2 each time. Then write n, n+2, n+4. For consecutive odd integers, write 3, 5, 7. Same gap of 2. Then write n, n+2, n+4. Both types use +2. Only plain consecutive integers (with no even or odd restriction) use +1: n, n+1, n+2.
Always test these four values in this order: a positive integer (try 2), a negative integer (try −2), zero, and one. These are the Four-Case Test values. The ACT designs its wrong answers to be true for positive integers but to fail for at least one of the other three. Testing only positive integers will cause you to incorrectly confirm wrong answer choices as “must be true.”
If the problem specifies “n is a positive integer,” skip the negative and zero cases but still test n=1 separately from n=2 or higher, since 1 behaves differently in expressions involving factors, exponents, and products. If the problem says “n is an integer” with no restriction, all four cases apply including zero and negatives.
The definition of a prime number requires exactly two distinct positive divisors: 1 and the number itself. The number 1 has only one positive divisor — itself — because 1 ÷ 1 = 1, and there is no other whole number that divides 1. Since 1 does not have two distinct positive divisors, it fails the definition and is not prime.
This is not an arbitrary rule. It matters mathematically because every integer has a unique prime factorization, and allowing 1 to be prime would break that property (you could multiply any factorization by 1 infinitely and get different “prime factorizations” of the same number). On the ACT, the test exploits this by placing 1 in lists labeled “only prime numbers.” Any list that includes 1 is automatically wrong when the question asks for only primes.
Yes. If the problem does not explicitly say the variable is an integer, decimals and fractions are valid values. This matters most on “must be true” questions: if n is not restricted to integers, you should also test n = 0.5 and n = −0.5 as test cases, because many statements that are true for all integers are false for fractions between 0 and 1.
Example: “n² > n” is true for all integers except 0 and 1. But if n = 0.5 (not an integer), then 0.25 > 0.5 is false. The statement fails for non-integer values. If the problem says “n is an integer,” you only test integers. If it says “n is a number” or does not specify, you must also test fractional values to disprove statements that only work for integers.