Decision and conditions
Use parity restrictions, inverse divisor counts and prime-exponent lower bounds. Prime factorization is unique; exponent zero is allowed in a divisor.
Worked example
How many positive odd divisors does 2^3 × 3^3 have? An odd divisor cannot contain any factor of 2. The exponent of 3 can be any integer from 0 through 3. There are 4 choices. Method choice: Use oddness to restrict one prime exponent before counting the other.
Contrast: change the reasoning
The square of a positive integer n is divisible by 2^7 × 3³. What is the least possible n? If n contains 2^u and 3^v, its square contains 2^(2u) and 3^(2v). Thus u≥4 and v≥2; exponents must be integers. The least n is 2^4×3²=144. Method choice: Double the unknown prime exponents, then find the smallest integer exponents satisfying both lower bounds.
Transfer with solution
Two positive integers have greatest common divisor 3 and least common multiple 108. One integer is 12. What is the other? For positive integers, their product equals gcd times lcm. 12b=3×108. Divide by 12: b=27. Method choice: Relate the product of the two integers to the product of their greatest common divisor and least common multiple.
Boundary check
For a nonnegative integer a, the integer N=2^a × 3² has exactly 12 positive divisors. What is a? A divisor has 4 choices for its power of 2 only after the total count is divided by the three choices for the power of 3. (a+1)×3=12. a+1=4, so a=3. Method choice: Count the choices for each prime exponent, including zero, and invert the resulting product.
Misconception to reject
Divisor count is not the sum of prime exponents. Counts all positive divisors without excluding even ones.
How many positive odd divisors does 2^3 × 3^3 have?
- An odd divisor cannot contain any factor of 2.
- The exponent of 3 can be any integer from 0 through 3.
- There are 4 choices.
Compare the other answer choices
Excludes exponent zero and misses the divisor 1.
Reports the power of 2 rather than counting allowed divisors.
Counts all positive divisors without excluding even ones.
Reports the largest odd divisor instead of the number of odd divisors.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GMAT exam content