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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
What is $\operatorname{lcm}(20,36)$ ?
- A.144
- B.4
- C.180
- D.200
- E.720
2.
Arithmetic sequence: first term $3$ , common difference $4$ . Sum of first $25$ terms?
- A.1300
- B.75
- C.99
- D.1275
- E.1271
3.
What is $\operatorname{lcm}(18,42)$ ?
- A.126
- B.144
- C.756
- D.84
- E.6
4.
How many positive integers $n\le 150$ are divisible by $6$ ?
- A.21
- B.26
- C.144
- D.24
- E.25
5.
Compute $2^{10}\bmod 7$ .
- A.3
- B.4
- C.5
- D.2
- E.2'
6.
Right triangle with legs $18$ and $24$ . Hypotenuse?
- A.42
- B.30
- C.29
- D.31
- E.24
7.
Evaluate $\displaystyle\sum_{k=1}^{50}\left\lfloor\frac{50}{k}\right\rfloor$ .
- A.203
- B.209
- C.2500
- D.100
- E.207
8.
What is $\operatorname{lcm}(33,45)$ ?
- A.528
- B.495
- C.450
- D.1485
- E.3
9.
An item costs $55$ . It is discounted $10\%$ , then $10\%$ tax is applied to the discounted price. How much is paid?
- A.61
- B.56.45
- C.54.45
- D.52.95
- E.50
10.
What is $\operatorname{lcm}(33,51)$ ?
- A.594
- B.561
- C.510
- D.1683
- E.3

Answer key · worked solutions

1. C. 180
$\operatorname{lcm}=ab/\gcd$ . $\gcd=4$ . $180$ .
2. D. 1275
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $1275$ .
3. A. 126
$\operatorname{lcm}=ab/\gcd$ . $\gcd=6$ . $126$ .
4. E. 25
Count multiples. $\lfloor 150/6\rfloor$ . $25$ .
5. D. 2
Successive squaring / cycles. Reduce powers modulo 7. $2$ .
6. B. 30
Pythagorean theorem / scaled triples. $\sqrt{(18)^2+(24)^2}$ . $30$ .
7. E. 207
Group equal floor values. Dirichlet blocks for $N=50$ . $207$ .
8. B. 495
$\operatorname{lcm}(a,b)=ab/\gcd(a,b)$ . $\gcd=3$ . $495$ .
9. C. 54.45
Discount first, then tax. $55\cdot(1-10/100)\cdot(1+10/100)$ . $54.45$ .
10. B. 561
$\operatorname{lcm}(a,b)=ab/\gcd(a,b)$ . $\gcd=3$ . $561$ .

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