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Mathbound · Contest practice

Mixed topics worksheet

Level 2

Name

Date

1.
Compute $3^{10}\bmod 7$ .
- A.6
- B.0
- C.4
- D.3
- E.5
2.
Compute $3^{5}\bmod 7$ .
- A.3
- B.6
- C.0
- D.1
- E.5
3.
What is the maximum product of positive integers that sum to $34$ ?
- A.131072
- B.177147
- C.236196
- D.236194
- E.236197
4.
What is $\operatorname{lcm}(14,42)$ ?
- A.42
- B.56
- C.588
- D.0
- E.14
5.
Compute $6^{5}\bmod 13$ .
- A.3
- B.4
- C.5
- D.2
- E.6
6.
Positive integer solutions to $x_1+\cdots+x_{4}=13$ ?
- A.221
- B.220
- C.216
- D.560
- E.715
7.
How many trailing zeros does $50!$ have in base $10$ ?
- A.11
- B.12
- C.10
- D.13
- E.14
8.
How many positive integers $n\le 150$ are divisible by $8$ ?
- A.19
- B.142
- C.17
- D.18
- E.16
9.
Arithmetic sequence: first term $3$ , common difference $3$ . Sum of first $15$ terms?
- A.375
- B.45
- C.360
- D.357
- E.z4
10.
Arithmetic sequence: first term $3$ , common difference $3$ . Sum of first $12$ terms?
- A.234
- B.231
- C.246
- D.36
- E.z4

Answer key · worked solutions

1. C. 4
Successive squaring / cycles. Reduce powers modulo 7. $4$ .
2. E. 5
Successive squaring / cycles. Reduce powers modulo 7. $5$ .
3. C. 236196
Prefer parts equal to $3$ (use $2$ s when needed; avoid $1$ s). Partition $34$ optimally. $236196$ .
4. A. 42
$\operatorname{lcm}=ab/\gcd$ . $\gcd=14$ . $42$ .
5. D. 2
Successive squaring / cycles. Reduce powers modulo 13. $2$ .
6. B. 220
$\binom{N-1}{k-1}$ . $\binom{12}{3}$ . $220$ .
7. B. 12
Exponent of $5$ in $n!$ . Sum $\lfloor n/5\rfloor+\lfloor n/25\rfloor+\cdots$ . $12$ .
8. D. 18
Count multiples. $\lfloor 150/8\rfloor$ . $18$ .
9. C. 360
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $360$ .
10. A. 234
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $234$ .

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