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Mathbound · Contest practice

Mixed topics worksheet

Level 2

Name

Date

1.
Compute $4^{5}\bmod 7$ .
- A.3
- B.4
- C.5
- D.2
- E.alt4
2.
Compute $2^{12}\bmod 11$ .
- A.6
- B.7
- C.4
- D.2
- E.5
3.
What is $\operatorname{lcm}(10,22)$ ?
- A.2
- B.110
- C.120
- D.220
- E.88
4.
How many positive integers $n\le 100$ satisfy $n\equiv 4\pmod{6}$ ?
- A.16
- B.17
- C.18
- D.19
- E.z4
5.
Compute $8^{10}\bmod 13$ .
- A.8
- B.0
- C.1
- D.2
- E.12
6.
What is $\operatorname{lcm}(18,22)$ ?
- A.216
- B.396
- C.176
- D.2
- E.198
7.
How many positive integers $n\le 50$ satisfy $n\equiv 3\pmod{4}$ ?
- A.11
- B.12
- C.14
- D.13
- E.15
8.
How many positive integers $n\le 50$ satisfy $n\equiv 1\pmod{8}$ ?
- A.8
- B.9
- C.6
- D.7
- E.z4
9.
How many positive integers $n\le 50$ are divisible by $9$ ?
- A.5
- B.41
- C.6
- D.8
- E.4
10.
What is $\operatorname{lcm}(12,60)$ ?
- A.72
- B.720
- C.0
- D.12
- E.60

Answer key · worked solutions

1. D. 2
Successive squaring / cycles. Reduce powers modulo 7. $2$ .
2. C. 4
Successive squaring / cycles. Reduce powers modulo 11. $4$ .
3. B. 110
$\operatorname{lcm}=ab/\gcd$ . $\gcd=2$ . $110$ .
4. B. 17
Arithmetic progression. First 4, step 6. $17$ .
5. E. 12
Successive squaring / cycles. Reduce powers modulo 13. $12$ .
6. E. 198
$\operatorname{lcm}=ab/\gcd$ . $\gcd=2$ . $198$ .
7. B. 12
Arithmetic progression. First 3, step 4. $12$ .
8. D. 7
Arithmetic progression. First 1, step 8. $7$ .
9. A. 5
Count multiples. $\lfloor 50/9\rfloor$ . $5$ .
10. E. 60
$\operatorname{lcm}=ab/\gcd$ . $\gcd=12$ . $60$ .

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Free AMC Math Worksheet Generator — Mathbound