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WUCT121
Discrete Mathematics
Numbers
Tutorial Exercises
1.
Natural Numbers
2.
Integers and Real Numbers
3.
The Principle of Mathematical Induction
4.
Elementary Number Theory
5.
Congruence Arithmetic
WUCT121
Numbers Tutorial Exercises
1
Section 1: Natural Numbers
Question1
What is a closed operation?
Question2
What are the closed operations on Õ?
Question3
What operations have an identity in Õ and what are they?
Question4
For those operations which have an identity, which elements of Õ have
inverses and what are they?
Question5
For the following, state which property is being demonstrated:
(a) a + b = b + a , a , b ∈  .
(b) 4( x + y ) = 4 x + 4 y,
x, y ∈  .
(c) 5 × (4 × 3) = (5 × 4) × 3 = 20 × 3 = 60 .
Question6
Simplify the following, indicating which property you are using at each
step: 8( x + y ) + ( y + x )15 .
Question7
State the Law of Trichotomy.
Question8
Use the definition of “<” to show 12 < 25.
Question9
What is the property of transitivity?
Question10
Show if a < b, then ac < bc where a , b, c ∈  .
Question11
State the well-ordering property for Õ.
WUCT121
Numbers Tutorial Exercises
2
Section 2: Integers and Real Numbers
Question1
What are the closed operations on ?
Question2
What operations have an identity in  and what are they?
Question3
For those operations which have an identity, which elements of  have
inverses?
Question4
Is  well-ordered? Demonstrate why or why not
Question5
Use the definition of odd to show if –67 is odd
Question6
What is a prime number?
Question7
What is a composite number?
Question8
What is a rational number?
Question9
What are the closed operations on ?
Question10
Which operations on  have identities and what are they?
Question11
For those operations which have an identity, which elements of  have
inverses?
Question12
WUCT121
Is  well-ordered? Demonstrate why or why not.
Numbers Tutorial Exercises
3
Section 3: The Principle of Mathematical Induction
Use Mathematical Induction to prove the following:
Question1
For all n ∈  , ( n + 1)!≥ 2n
n −1
Question2
For all n ∈  , cos x × cos 2 x × cos 4 x × L × cos 2
Question3
∀n ∈ ,12 + 2 2 + L + n 2 =
Question4
∀n ∈ , ∃z ∈  : 4 n − 1 = 3z
Question5
∀n ∈ , n 3 ≥ 5n − 4
Question6
Prove that for every real number x and each n ∈ , (1 + x ) n ≥ 1 + nx .
Question7
Prove that 2 + 6 + 10 + K + ( 4n − 2) = 2n 2 for all n ∈  .
Question8
Prove that 7 n − 2 n is divisible by 5 for all n ∈  .
Question9
Find the mistake in the following proof fragment.
Claim(l) is 12 =
2 n sin x
n(n + 1)(2n + 1)
6
Let Claim(n) be: 12 + 2 2 + L + n 2 =
Step 1:
x=
sin 2 n x
n(n + 1)(2n + 1)
for all n ∈  .
6
1(1 + 1)(2 × 1 + 1)
6
LHS = 1, RHS = 1, so LHS = RHS.
Therefore, Claim(1) is true.
Step 2:
Assume that Claim(k) is true for some k ∈ 
k (k + 1)(2k + 1)
that is, k 2 =
6
(1)
Prove Claim(k + 1) is true;
that is, prove that (k + 1)2 =
WUCT121
(k + 1)((k + 1) + 1)(2(k + 1) + 1)
Numbers Tutorial Exercises
6
4
Question10
Let Claim(n) be: For all integers n ≥ 1, 3n − 2 is even.
What is wrong with the following proof ?
Suppose the Claim(k) is true for some k ∈  . That is, suppose that 3k − 2 is even.
We must show that Claim(k + 1) is true.
That is, we must show that 3k +1 − 2 is even.
Now,
3k +1 − 2 = 3 × 3k − 2
= (1 + 2) × 3k − 2
= 2 × 3k + 3k − 2
Now 3k − 2 is even by the hypothesis and 2 × 3k is even by inspection. Hence, the
sum of the two quantities is even.
Therefore, Claim(k + 1) is true.
Therefore, Claim(n) is true for all n ∈  .
Question11
Let Claim(n) be the statement “ n 2 + n + 11 is a prime number”.
(a)
Verify that Claim(n) is true for n = 1, 2, 3, 4, 5, 6, 7, 8, and 9.
(b)
Can we conclude that Claim(n) is true for all n ∈  ? Explain.
Question12
Try to use induction to prove that 1 +
(a)
1 1
1
+ +K+
≤ 2 for all n ∈  .
2 4
2n
What goes wrong?
Prove that 1 +
(b)
1 1
1
1
+ +K+
≤2−
for all n ∈  .
n
2 4
2
2n
thus, showing that 1 +
1 1
1
+ +K+
≤ 2 for all n ∈  .
2 4
2n
Question13
1⎞ ⎛
⎛ 1 ⎞⎛
Prove that ⎜1 + ⎟⎜1 + ⎟ K ⎜1 +
2⎠ ⎝
⎝ 1 ⎠⎝
Question14
Prove ∑ 2 i = 2(2 n − 1)
1⎞
⎟ = n + 1 for all n ∈  .
n⎠
n
WUCT121
i =1
Numbers Tutorial Exercises
5
n
1
n
=
i =1( 4i − 3)(4i + 1) 4n + 1
Question15
Prove ∑
Question16
Prove n3 − 4n + 6 is divisible by 3 for all n ∈ 
Question17
Use the identity ∑ i =
Question18
Prove ∑ (2i − 1)2 =
Question19
Prove ∑ i (i + 2) =
Question20
Prove 1 + 3 + K + (2n − 1) = n 2 for all n ∈  .
Question21
If bn = 3bn −1 , n ≥ 2, b1 = 3 . Prove bn = 3n for all n ∈  .
Question22
If un = un −1 + n − 1, n ≥ 2, u1 = 0. Prove un =
Question23
If un = 2un −1 + 1, n ≥ 2, u1 = 1. Prove un = 2 n − 1 ∀n ∈ .
Question24
If sn = ( −n) sn −1 , n ≥ 2, s1 = −1 . Prove sn = ( −1) n n! , ∀n ∈ .
Question25
If un = 5un −1 − 6un − 2 , n ≥ 3, u1 = 2, u2 = 4 . Prove un = 2 n , ∀n ∈ .
Question26
If un = 5un −1 − 6un − 2 , n ≥ 3, u1 = 1, u2 = 5 . Prove un = 3n − 2 n , ∀n ∈ .
n
i =1
n
i =1
n
i =1
2
n
⎛ n ⎞
n( n + 1)
to prove that ∑ i 3 = ⎜⎜ ∑ i ⎟⎟
2
i =1
⎝ i =1 ⎠
n(2n − 1)(2n + 1)
3
n( n + 1)(2n + 7)
6
n( n − 1)
, ∀n ∈ .
2
Question27 Let u1 , u2 ,..., un ... be real numbers such that
un = −2un −1 + 3un − 2 , n ≥ 3, u1 = 2, u2 = 2 . Prove that un = 2 for all n ∈  .
Question28
WUCT121
If an = 2an −1 − an − 2 , n ≥ 3, a1 = 3, a2 = 5. Prove an = n + 2, ∀n ∈ . .
Numbers Tutorial Exercises
6
Section 4: Elementary Number Theory
Question1
(a)
Let m, n ∈ . Is 6m + 8n even? Why?
(b)
Let r , s ∈ . Is 6r + 4s 2 + 3 odd? Why?
(c)
Let u, v ∈ , with u > v. Is u 2 − v 2 composite? Why?
Question2
Write down the definition of divisibility.
Question3
(a) Is 52 divisible by 13? Why?
(b) Let k ∈ . Is 3 a divisor of (3k + 1)(3k + 2)(3k + 3) ? Why?
(c) Does 7 | 13 ? Why?
(d) Let a, b ∈ . Does 4 | (6a × 10b) ? Why?
Question4
State the Quotient Remainder Theorem.
Question5
State the Fundamental Theorem of Arithmetic.
Question6
Use a sieve of Eratosthenes to find the primes between 201 and 300
Question7
List the twin primes between 201 and 300.
Question8
Find the prime factorization, gcd and lcm for the following pairs of
numbers:
(a) 369, 8619
(c) 1375, 605
(b) 936, 7623
(d) 4968, 9000
Use the Euclidean Algorithm to find the gcd(a, b) for the following pairs
of numbers. Find values m, n ∈ . such that gcd(a, b) = am + bn
Question9
(a) a = 72, b = 63.
(e) a = 63, b = 24.
(b) a = 2104, b = 21.
(f) a = 336, b = 60.
(c) a = 15, b = 10.
(g) a = 7684, b = 4148.
(d) a = 9, b = 5.
(h) a = 90, b = –54.
WUCT121
Numbers Tutorial Exercises
7
Question10
Give three pairs of numbers that are relatively prime.
Question11
Fermat Primes are given by the formula F (n) = 22 + 1, for n ∈  .
n
(a) Write down the first three Fermat Primes. Are they all prime?
(b) Find F(4) and F(5). Are they prime?
Question12
Mersenne Primes are given by the formula g (n) = 2n − 1, for n ∈  .
(a) Find g(n) for 2, 3, 4, 5, 6, 7, 8, 9.
(b) Which of the values for g(n) found in part (a) are prime?
(c) For what values of n do you suggest that g(n) will give a prime number?
(d) Find g(11). How does this compare with your opinion for part (c)?
(e) Prove that whenever n is composite, then g(n) is also composite. [Hint: Let
n = pq, for some p, q ∈  , and factorise g(n).]
Question13 What can you say about 3n − 1, for n ∈  ? Will this produce prime
numbers? Why or why not?
Question14 Show that in any group of 367 people at least two people must have the
same birthday. Does it work for 366 people?
Question15 Seven different integers are chosen from 1 to 12 inclusive. Must at least
one be odd?
Question16 Let T = {1, 2, 3, 4, 5, 6, 7, 8, 9} . Five integers are chosen from T. Must there
be two which add to 10?
Question17 A programmer writes 500 lines of computer code in 17 days. Must there
be at least one day the programmer wrote 30 or more lines of code?
Question18 Show that at a party of 10 people at least two of the people must have the
same number of friends at the party. You may assume friendship is commutative, that
is A is friend of B ⇔ B is friend of A.
WUCT121
Numbers Tutorial Exercises
8
Section 5: Congruence Arithmetic
Question1
(a) Find x ∈ , 0 ≤ x < 11 so that 129 ≡ x(mod11) .
(b) Find x ∈ , 0 ≤ x < 13 so that 311 ≡ x(mod13) .
(c) Find x ∈ , − 11 < x ≤ 0 so that 4 ≡ x(mod11) .
Question2
Use the definition of congruence modulo to prove the following properties
for congruence modulo 3.
(a) ∀a ∈ , a ≡ a (mod 3) .
(Reflexive Property)
(b) ∀a, b ∈ , a ≡ b(mod 3) ⇒ b ≡ a(mod 3)
(Symmetric Property)
(c) ∀a, b, c ∈ , a ≡ b(mod 3) ∧ b ≡ c(mod 3) ⇒ a ≡ c(mod 3)
Question3
(Transitive Property)
Let n = 7 .
(a) For m ∈ , is [m] (in 7 ) a number, a set of numbers or a set of sets?
(b) Describe [2], [5] and [7] in 7 by listing eight elements in each set. Write down
the set 7 .
(c) True or false (in 7 ) ?
(i) [2]=[9]
(iv) 3 ∈7
(ii) [5] ≡ 5(mod 7 )
(v) [10] ∈7
(iii) 5 ∈ [19]
Question4
(vi) a ≡ b(mod 7 ) ⇒ [ a ] ∈ [b]
Write out the addition and multiplication tables for 5 , 6 and 1 .
(a) Do all elements in 5 have multiplicative inverses?
(b) Do all elements in 6 have multiplicative inverses?
(c) Let [m] ∈6 have an inverse under multiplication, What can you say about
gcd(m, 6)
Question5
WUCT121
Let n ∈ . Prove that [a ] = [b] ⇔ a ≡ b(mod n )
Numbers Tutorial Exercises
9
Question6
In 11 ,
(a) Find values of m and n such that 410 ∈ [m] and 411 ∈ [n] .
Hint: Find m ∈ , 0 ≤ m < 11 so that 410 ≡ m(mod11)
(b) Find values of i and j such that 310 ∈ [i ] and 311 ∈ [ j ] .
(c) Based on your results from part (a) and (b), can you make a conjecture about
x10 (mod11) and x11(mod11) , where x ∈  and 0 < x < 11 ?
Question7
Let n = 9 .
(a) Write out the addition and multiplication tables for 9 .
(b) True or false (in 9 ) ?
(i) [3]=[–13]
(v) The element [1] is the identity of
9 under addition
(ii) − 9 ∈ [19]
(vi) Every element of 9 has a
(iii) 2 ∈9
(iv) 4 ≡ 9(mod 5) ⇒ 4 ∈ [5] (in 9 )
Question8
multiplicative inverse.
Every natural number m can be written in the form
m = d k × 10k + d k −1 × 10k −1 + K + d 2 × 100 + d1 × 10 + d 0 , where di ∈ . is the digit
in the 10ith position of m.
7526 = 7 × 1000 + 5 × 100 + 2 × 10 + 6
Examples :
57402 = 5 × 10000 + 7 × 1000 + 4 × 100 + 0 × 10 + 2
(a) Let S be the sum of the digits of 7526. Find S.
Find x in each case, where 0 ≤ x < 9
(i) 7 × 1000 ≡ x(mod 9 )
(ii) 5 × 100 ≡ x(mod 9 )
(iii) 2 × 10 ≡ x(mod 9 )
(iv) (7 × 1000 + 5 × 100 + 2 × 10 + 6) ≡ x (mod 9 )
Deduce that 7526 ≡ S (mod 9 ) .
WUCT121
Numbers Tutorial Exercises
10
(b) Let S be the sum of digits of 57402. Show that 57402 ≡ S (mod 3) .
(c) Prove that for any integer m, if the sum of the digits of m is S, then m ≡ S (mod 3)
and m ≡ S (mod 9 ) .
Hint: Use the fact that 10n ≡ 1(mod 3) and 10n ≡ 1(mod 9)
(d) What divisibility rules have just been shown?
WUCT121
Numbers Tutorial Exercises
11