Decision and conditions
Count choices of prime exponents and distinguish a perfect square from a multiple. Factor the integer completely. Zero is an allowed exponent in a divisor. A square requires every exponent to be even.
Worked model
How many positive divisors does 2^3 × 3^2 have? The exponent of 2 can be any of 0 through 3: 4 choices. The exponent of 3 has three choices: 0, 1, 2. Multiply the independent counts to obtain 12. Before calculating: A divisor independently chooses an exponent for each prime.
Contrast: a different decision
What is the least positive integer m for which 3456m is a perfect square? 3456 = 2^7 × 3³. Both prime exponents are odd, so each needs one additional factor. The least multiplier is 2 × 3 = 6. Before calculating: Which prime exponents need one more factor to become even?
Transfer with solution
How many trailing zeros are in 40! (the product of the positive integers from 1 to 40)? A trailing zero consumes one factor 2 and one factor 5; factors of 2 are plentiful. Multiples of 5 contribute 8, with 1 additional fives from multiples of 25. The total is 9. Before calculating: Count factors of five, including the extra five in each multiple of 25.
Check the boundary
Adding exponents counts prime factors with multiplicity, not divisors. Two integers are 2^3 × 3² and 2² × 3^4. What is their least common multiple? The larger exponent of 2 is 3. The larger exponent of 3 is 4. The least common multiple is 2^3 × 3^4 = 648.
How many positive divisors does 2^3 × 3^2 have?
- The exponent of 2 can be any of 0 through 3: 4 choices.
- The exponent of 3 has three choices: 0, 1, 2.
- Multiply the independent counts to obtain 12.
Compare the other answer choices
Counts prime factors with multiplicity rather than divisors.
Multiplies exponents, omitting exponent zero from each choice list.
Allows only exponents zero and one for prime 3, omitting exponent two.
Reports the integer itself rather than counting its 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.
GRE Quantitative Reasoning overview