Read a Number by Its Prime Exponents
Write each number as a product of prime powers and the question becomes a question about exponents. A perfect square has every exponent even, the divisor count multiplies (exponent + 1) over the primes, and a fraction terminates exactly when its reduced denominator has no prime but 2 and 5. Grade-school divisibility tested one divisor at a time; this reads the whole factor structure at once.
grade 6–8 Number systemExpressions Identify SubproblemsMake a Systematic List
Progression (4)
g6 1. Smallest square that is a common multiple Number systemExpressions
g6 2. Fix primes by sum, exponents by divisor count OperationsExpressions
g8 3. Make every prime exponent even Number systemOperations
g8 4. Count numerators that cancel the denominator's 3s Number systemOperations