Number Properties
Recognize the structure of an integer before doing unnecessary computation.
Divisibility, Factors, Multiples, and Remainders
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Number properties reveal what an integer must be able to do
Divisibility rules test whether division leaves no remainder. Factors divide a number exactly; multiples are produced by multiplying it by integers.
a = dq + r, where 0 ≤ r < dPrime factorization makes GCF, LCM, factor counting, perfect squares, and many remainder questions easier to see.
GCF versus LCM
Use GCF when dividing quantities into the greatest number of identical groups. Use LCM when cycles must meet again.
Remainders have a limit
A remainder is always smaller than the positive divisor. A remainder of 7 is impossible if the divisor is 7 or less.
Translate the wording before calculating
Greatest identical grouping → GCF. First time cycles coincide → LCM. Leaves r when divided by d → n = dq + r. For divisibility, test the smallest useful rule first.
Why it works
The wording tells you whether you need a common divisor, a common multiple, or a remainder relationship. Naming that structure prevents trial-and-error computation.
Five forms you should recognize
Problem: The number 42,57□ is divisible by 5 and 9. Find □.
Last digit: 0 or 5; digit sum: 18+□Why: Only 0 satisfies both divisibility conditions.
Problem: Divide 84 red and 126 blue flags into the greatest number of identical sets.
GCF(84,126) = 42Why: The number of sets must divide both quantities exactly.
Problem: Lights flash every 18 and 24 seconds. When do they meet again?
LCM(18,24) = 72 secondsWhy: Seventy-two is the first positive time belonging to both schedules.
Problem: n leaves remainder 5 when divided by 7. Find the remainder of n + 10.
5 + 10 = 15; 15 = 2(7) + 1Why: Only the leftover part matters, and any total remainder at least 7 is reduced again.
Problem: Find the least multiplier that makes 540 a perfect square.
540 = 2²×3³×5 → multiply by 3×5 = 15Why: Every prime in a perfect square must have an even exponent.
Check before you commit
- Using LCM when a problem asks for the greatest number of equal groups
- Multiplying cycle lengths instead of finding their least common multiple
- Allowing a remainder equal to or greater than the divisor
- Assuming a number is prime because it is odd
- Forgetting to subtract an announced common remainder before finding a GCF
- Adding prime exponents instead of using the divisor-count formula
Do you need the lesson-or just practice?
One original question in each form recommends your next step. It does not yet verify mastery.
Work at the level you need.
Foundations
Build the core procedure with immediate explanations.
Core Practice
Use mixed forms with less scaffolding.
UPCAT-Style Transfer
Apply the competency in unfamiliar representations.
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A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.
Number Properties FAQ
How do I decide between GCF and LCM?
GCF divides existing quantities into the largest identical units; LCM finds the earliest shared multiple or meeting time.
How far must I test to decide whether a number is prime?
Test prime divisors only up to the square root of the number.
How do I count the factors of a number?
If n = pᵃqᵇ, then it has (a + 1)(b + 1) positive factors because each prime exponent can range from zero to its maximum.
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