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Answer key · worked solutions

1. 1. D. 165

   Per prime exponent $e$: $2e+1$ pairs. Multiply across primes. $165$.

2. 2. A. 92

   For each $k$, count $n\in[k^{2},(k+1)^{2})$ divisible by $k$. Careful floor enumeration. $92$.

3. 3. D. 9

   Exponent of $5$ in $n!$. Sum $\lfloor n/5\rfloor+\lfloor n/25\rfloor+\cdots$. $9$.

4. 4. B. 508

   Sieve marking $\omega(n)=2$ where $\omega$ counts distinct primes. Mark multiples of primes; count $n$ hit exactly twice. $508$.

5. 5. E. 12

   Exponent of $5$. $10+2$. $12$.

6. 6. D. 31

   Exponent of $5$ in $n!$. Sum $\lfloor n/5\rfloor+\lfloor n/25\rfloor+\cdots$. $31$.

7. 7. E. 180

   Max exponents. $2^2\cdot3^2\cdot5$. $180$.

8. 8. C. 455

   For each prime with LCM exponent $e$, there are $2e+1$ exponent pairs. Multiply over primes. $455$.

9. 9. C. 105

   Per prime factor of $360=2^3\cdot3^2\cdot5$, $\max$ exponents give $2e+1$ choices. $(2\cdot3+1)(2\cdot2+1)(2\cdot1+1)=7\cdot5\cdot3$. $105$.

10. 10. E. 105

    For each prime with LCM exponent $e$, there are $2e+1$ exponent pairs. Multiply over primes. $105$.