Problem 22 · 2023 Math Kangaroo
Stretch
Number Theory
divisibilityfactorization
How many positive integers divide \(2^{20} \cdot 3^{23}\) but not \(2^{10} \cdot 3^{20}\)?
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Answer: C — 273
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Hint 1 of 2
Count divisors using the exponent-plus-one rule for each prime.
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Hint 2 of 2
Every divisor of 210·320 already divides 220·323, so subtract.
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Approach: count divisors of each and subtract the overlap
- 220·323 has (20+1)(23+1) = 504 divisors; 210·320 has (10+1)(20+1) = 231 divisors.
- Since 210·320 divides 220·323, all 231 of its divisors also divide the first number.
- Divisors of the first but not the second: 504 − 231 = 273.
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