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Mathbound · Contest practice

Divisibility worksheet

Mixed difficulty

Name

Date

1.
How many positive integers $n\le 100$ satisfy $\gcd(n,100)=1$ ? (That is, $n$ shares no prime factor with $100$ .)
- A.50
- B.40
- C.60
- D.20
- E.67
2.
How many pairs of positive integers $(x,y)$ with $x\le y$ satisfy $\dfrac1x+\dfrac1y=\dfrac1{12}$ ?
- A.9
- B.8
- C.12
- D.6
- E.7
3.
How many positive integers $n\le 100$ have $\gcd(n,100)=1$ ?
- A.50
- B.40
- C.60
- D.20
- E.67
4.
How many positive divisors does $43$ have?
- A.3
- B.4
- C.1
- D.2
- E.5
5.
How many positive divisors does $11$ have?
- A.2
- B.4
- C.3
- D.11
- E.1
6.
What is $\operatorname{lcm}(12,24)$ ?
- A.36
- B.24
- C.0
- D.288
- E.12
7.
What is $\operatorname{lcm}(27,51)$ ?
- A.486
- B.459
- C.408
- D.1377
- E.3
8.
How many positive integers $k\le 60$ satisfy $\gcd(k,60)=1$ ?
- A.8
- B.26
- C.59
- D.30
- E.16
9.
What is $\operatorname{lcm}(12,30)$ ?
- A.360
- B.6
- C.72
- D.60
- E.30
10.
What is $\operatorname{lcm}(20,56)$ ?
- A.1120
- B.224
- C.4
- D.280
- E.300

Answer key · worked solutions

1. B. 40
Euler totient $\varphi(100)$ . $100(1-\frac12)(1-\frac15)=40$ . $40$ .
2. B. 8
$(x-12)(y-12)=144$ . Divisors $d\le12$ of $144$ . 8
3. B. 40
$\varphi(100)$ . $100(1-1/2)(1-1/5)=40$ . $40$ .
4. D. 2
Factor and use $\prod(e_i+1)$ . Count divisors of $43$ . $2$ .
5. A. 2
Factor and use $\prod(e_i+1)$ . Count divisors of $11$ . $2$ .
6. B. 24
$\operatorname{lcm}(a,b)=ab/\gcd(a,b)$ . $\gcd=12$ . $24$ .
7. B. 459
$\operatorname{lcm}(a,b)=ab/\gcd(a,b)$ . $\gcd=3$ . $459$ .
8. E. 16
Euler totient $\varphi(n)$ . Factor 60. $16$ .
9. D. 60
$\operatorname{lcm}(a,b)=ab/\gcd(a,b)$ . $\gcd=6$ . $60$ .
10. D. 280
$\operatorname{lcm}=ab/\gcd$ . $\gcd=4$ . $280$ .

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