Integer properties on ACT Math cover primes, even and odd rules, consecutive integers, and remainders—number-sense questions that reward quick reasoning instead of long algebra. They show up as standalone items and as the hidden logic inside harder word problems.
Primes and factors
A prime has exactly two factors: 1 and itself. Know the primes under 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Two is the only even prime. Factor questions often ask how many factors a number has—prime factorize first, then use exponents. If 72 = 2^3 times 3^2, the factor count is (3 + 1)(2 + 1) = 12.
Even and odd
- Even ± even = even; odd ± odd = even; even ± odd = odd
- Even times anything = even; odd times odd = odd
- Any even power of an odd number is odd; any power of an even number is even (except 0^0, which you will not see)
If a question asks whether n squared minus n is even, factor: n(n - 1) is the product of two consecutive integers, so one is always even. The product is always even. That kind of structural argument beats plugging in every time.
Consecutive integers
Three consecutive integers can be written as n, n + 1, n + 2—or sometimes n - 1, n, n + 1 to center on zero. Their sum is always 3n + 3, which is divisible by 3. Consecutive even integers step by 2: n, n + 2, n + 4. Translate the words, then use sum or product formulas.
Remainders and modular thinking
When a question asks for the remainder when a large number is divided by 7, look for a pattern or use known divisibility rules. What is the remainder when 4^100 is divided by 7? Powers of 4 mod 7 cycle: 4, 2, 1, 4, 2, 1. Since 100 is a multiple of 3, the remainder is 1. You do not need to compute 4^100. Remainder logic connects to exponent rules when powers repeat in cycles.
Where to practice
These topics sit in Pre-Algebra and Number & Quantity on the enhanced ACT. They are fast points if you know the rules cold. For the full map of what else appears, see ACT Math topics. Drill a few integer-property questions each study session—they compound quickly.
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