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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
An item costs $55$ . It is discounted $15\%$ , then $10\%$ tax is applied to the discounted price. How much is paid?
- A.49.93
- B.47
- C.61
- D.53.43
- E.51.43
2.
What is $\operatorname{lcm}(18,62)$ ?
- A.2
- B.558
- C.576
- D.1116
- E.496
3.
An item costs $32$ . After a $25\%$ discount (no tax), how much is paid?
- A.24
- B.7
- C.32
- D.25.00
- E.8
4.
What is $\operatorname{lcm}(12,68)$ ?
- A.204
- B.216
- C.816
- D.136
- E.4
5.
If $r$ and $s$ are roots of $x^2+9x+8=0$ , find $r^3+s^3$ (without solving).
- A.72
- B.729
- C.513
- D.505
- E.522
6.
How many prime numbers satisfy $20 < p < 40$ ?
- A.6
- B.4
- C.7
- D.3
- E.5
7.
How many positive integers $k\le 200$ have exactly three positive divisors?
- A.5
- B.6
- C.14
- D.7
- E.9
8.
How many positive divisors does $43$ have?
- A.2
- B.4
- C.3
- D.43
- E.1
9.
Arithmetic sequence: first term $4$ , common difference $2$ . Sum of first $15$ terms?
- A.285
- B.60
- C.32
- D.270
- E.268
10.
How many positive integers $n\le 50$ satisfy $n\equiv 2\pmod{7}$ ?
- A.8
- B.10
- C.6
- D.7
- E.9

Answer key · worked solutions

1. E. 51.43
Discount first, then tax. $55\cdot(1-15/100)\cdot(1+10/100)$ . $51.43$ .
2. B. 558
$\operatorname{lcm}=ab/\gcd$ . $\gcd=2$ . $558$ .
3. A. 24
Multiply by $(1-d)$ . $32(1-25/100)$ . $24$ .
4. A. 204
$\operatorname{lcm}=ab/\gcd$ . $\gcd=4$ . $204$ .
5. C. 513
$r^3+s^3=(r+s)((r+s)^2-3rs)$ . Vieta: $r+s=9$ , $rs=8$ . $513$ .
6. B. 4
List. $23,29,31,37$ . $4$ .
7. B. 6
Exactly three divisors $\iff k=p^2$ . Primes $p\le\sqrt{200}$ . $6$ .
8. A. 2
Factor and use $\prod(e_i+1)$ . Count divisors of $43$ . $2$ .
9. D. 270
$S_n=\frac n2(2a+(n-1)d)$ . Plug values. $270$ .
10. D. 7
Arithmetic progression. First 2, step 7. $7$ .

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