Download MATH 13187: Freshman Seminar, spring 2016 Practice for Exam 1

MATH 13187: Freshman Seminar, spring 2016
Practice for Exam 1
Instructions: Clearly explain the answers to the following questions. If appropriate,
you may do this by indicating steps clearly without many words. Showing your work
may help you get partial credit. Note: this practice exam is about twice the length
of the exam you will take on Monday, February 22.
1. Suppose the olympic basketball coach has 1 center, 5 forwards, and 3 guards on
her team. Her starting line-up must have 1 center, 2 forwards, and 2 guards.
How many different starting line-ups can she choose?
1
5
3
Brief answer:
×
×
= 30.
1
2
2
2. (a) Find gcd(20, 48) and lcm(20, 48).
20 · 48
= 240.
4
(b) Is it possible to write 4 as a combination of 20 and 48? If yes, show a way
of doing it. If no, explain why not.
Brief answer: gcd(20, 48) = 4. lcm(20, 48) =
Brief answer: yes, because 4 is a multiple of gcd(20, 48). In fact, 4 = 3·48−7·20,
which you can find by listing multiples of 20 and 48.
(c) Is it possible to write 6 as a combination of 20 and 48? If yes, show a way
of doing it. If no, explain why not.
Brief answer: no, because 6 is not a multiple of gcd(20, 48).
3. (a) Give the prime factorization of 198.
Brief answer: 198 = 2 · 32 · 11.
(b) Decide whether 199 is prime.
Brief answer: 199 is prime because the primes less than
divide 199.
√
199 = 14.xxx do not
4. There are 4 toppings at the salad bar for you to put (or not put) on the bowl
of lettuce you hold in your hands.
(a) Assuming that you like every topping, how many possible salads could you
create with two different toppings, if order does not matter?
4
Brief answer:
.
2
5. How many numbers are there between 20 and 120 that are not divisible by 6?
How many numbers are there between 20 and 120 that are divisible by 6 but
not by 15?
Brief answer: 101 − 17 = 84 are not divisible by 6. 17 − 4 = 13 are divisible by
6 but not by 15.
6. (Unit 1 and 1.4 of Unit 3)
(a) How many numbers from 48 to 237 are multiples of 14?
Brief answer: 16 − 4 + 1 = 13
(b) How many numbers from 48 to 237 are multiples of 20?
Brief answer: 11 − 3 + 1 = 9
(c) How many numbers from 48 to 237 are common multiples of 14 and 20?
Brief answer: 1, it is same as multiples of 140
(d) How many numbers from 48 to 237 are multiples of 14 or 20?
Brief answer: 13 + 9 − 1 = 21.
(e) How many numbers from 48 to 237 are not multiples of 14 or 20?
Brief answer: 237 − 48 + 1 − 21 = 169.
7. Find lcm(271, 620). How does it compare to 271 × 620?
Brief answer: By the Euclidean algorithm gcd(271, 620) = 1, so lcm(271, 620) =
271 · 620 = 168020.
8. Write 1 as a combination of 11 and 29.
Brief answer: 1 = 8 · 11 − 3 · 29.
9. Write 1 as a combination of 351 and 854. Write 3 as a combination of 351 and
854.
Brief answer: 1 = 309 · 351 − 127 · 854. Multiply both sides by 3 to get 3 =
927 · 351 − 381 · 854.
10. Give a prime factorization of 844.
Brief answer: 844 = 22 · 211.
11. Find the smallest prime bigger than 300.
Brief answer: 307 (use the sieve of Eratosthenes from 300 to 320).
43
43
44
m
12. Express
+
+
as a binomial coefficient
.
13
14
13
k
45
Brief answer:
.
14
13. Suppose Cathy is choosing a committee of four kids from a class with 14 boys
and 16 girls.
(a) How many ways can she choose the committee?
30
Brief answer:
.
4
(b) How many ways can she choose the committee if it must have 2 boys and
2 girls?
14
16
Brief answer:
·
.
2
2
(c) How many ways can she choose the committee if it must have 3 boys and
one girl?
14
Brief answer:
· 16.
3
14. (Unit 4, 1.4) Seventeen people are running for the Republican nomination for
president. Fred is required to break them into two groups for debates. If the
first debate is to have 8 people, how many ways can Fred choose the people?
17
Brief answer:
.
8
15. (a) How many ways are there to pick four different numbers between 5 and 43
if the order in which they are chosen matters?
39!
Brief answer:
35!
(b) How many ways are there to pick four different numbers between 5 and 43
if the order in which they are chosen does not matter?
39
Brief answer:
.
4
16. (Unit 2)
(a) How many license plates are there with 3 letters and 4 numbers?
Brief answer: 263 · 104 .
(b) How many license plates are there with 3 letters and 4 numbers, if the last
letter cannot be X, and the first number cannot be 0?
Brief answer: 25 · 262 · 9 · 103 .
17. (Unit 9) How many numbers divide the number 26,000?
Brief answer: 5 · 4 · 2 = 40.
18. (Unit 9) How many numbers divide 211?
Brief answer: 2, just 1 and 211 since 211 is prime.
19. (Unit 9)(a) How many numbers divide 24 · 3 · 57 · 13?
Brief answer: 5 · 2 · 8 · 2 = 160.
(b) Let d = gcd(m, n), where m = 22 · 32 · 5 · 7 and n = 24 · 3 · 57 · 13. Find the
prime factorization of d. How many numbers divide d?
Brief answer: d = 22 · 3 · 5. d has 3 · 2 · 2 = 12 divisors.
(c) Find the prime factorization of the least common multiple of 22 · 32 · 5 · 7
and 24 · 3 · 57 · 13.
Brief answer: 24 · 32 · 57 · 7 · 13.
48
20. Does 35 divide
?
18
Brief answer: no. Since 35 = 5 · 7 is the prime factorization of 35, for 35 to
divide a number
n, both 5 and 7 must divide the number. Since 5 does
not
48
48
divide
(to see this, look for multiples of 5 in the expression for
),
18
18
48
35 does not divide
.
18
48
21. Does 19 divide
?
18
48
Brief answer: yes. In the expression for
, there is a multiple of 19 in the
18
48
numerator, but none in the denominators. This means
equals 19 times a
18
product of other primes, so it is divisible by 19.