程序代写代做代考 ER C MATH1061/7861, Thu 20 Aug 2020

MATH1061/7861, Thu 20 Aug 2020
Terminology: “Distinct” is another word for “different”.
Question 0. Which one of the following collections of values for a, b and m provides a counterexample that disproves the following statement?
∀a,b,m ∈ Z, if m|(a+b) then m|(a−b).
◦ a = 8, b = 8, m = 4
◦ a = 8, b = 16, m = 4
◦ a = 11, b = 3, m = 4 ◦a=7,b=5,m=4 mm
Question 1. Prove or disprove:
∀m ∈ Z, 6m(2m + 10m2) is divisible by 4.
Itai’s mta.br
t
fm
f Need to write n integer
Timon’s 6m 2 mt5m2
Of q.e.shwafnisai.at 4
2eOmClt5m 12 m2 I 15M
4 3 m2 I15M since 3m41 15M
ER
i 416M Gaeomy
a
6
4
Question 2. Prove or disprove: ∀a,b,c ∈ Z, if a|(b+c) then a|b or a|c.
Edsel Tes atbadat ata2 bcl
21111 but 2ft
a Ib
ME a ate
i albec but atb airdate Question 3. Prove or disprove:
∀a,b,c ∈Z, if a is a multiple of c, then ab is a multiple of c.
My Proof
use acy
use
Tiesto
211 MY
Totals
hard
to
want clas
ie.ab D
iab Sine kbc I
c
Kb
we have ab is a multipleofc I
a
k Chea
L.IE 4 E c.k.b
csitEaO

Theorem. √2 is irrational. Oort
Proof by contradiction Assume
3 4
i
i
p.ge q.to common factors creept Is
485,124 FL is rational E
where
Andy p q have
qg.gg
Pt 1
qq.pw.fm
num Eden have n
common
factors apt
2
Se
i i
772s
F
LI is is
p 2k for KEI i p24h2
i 292 467 d 92215
i 92 is
i
even
i q 2e for some Eee
pig hare 2 as a common factor
CONTRADICTION t
I
52
is
irrational
I
wet pus is even
p2
i pis
even
2Wh
is evenDd eatin
of odd
ah

Question 4. What is the unique prime factorisation of 6975, written in standard factored form (which means the primes appear
in increasing order)?
6975
I
Check 2
Tx
Question 5. How many positive divisors does 6975 have?
5 1395
275g 16 1bar
mm
27 9 5 5 3 93
E
5 5 3 3.31
3 5231
2O
divisors 332
2 3,91
Question 6. What is the unique prime factorisation of 69753, written in standard factored form? 18M
3 69753
g
x y3 a 3 oc
3252.31
y
Question 7. Prove, disprove, or salvage if possible: For any integers a and b, if a | b then a ∤(b+1).
323.52 3,3 True
Challenge A. Prove, disprove, or salvage if possible:
∀n ∈ N, the sum of any n consecutive integers is divisible by n.
Question 8. Find the smallest positive integer n such that 1400n is a perfect square. Perfectsquad All egsonents in primefeet have
1400 it a
Ansi
7
14
Need
x2 7
a 14 2 7 i
10 10 14 25 2.5 Z
23.52 7
1400N P.S 2352 76n
n
Try
2
24 52.72 5.79
Question 9 (Wednesday). Prove that the sum of any rational number and any irrational number is irrational.
Challenge B (Monday). Prove the following statement by contradiction: For all m ∈ N, if m, m + 2 and m + 4 are all prime, then m = 3.
False
Disprove
FIX THE THEOREM
3456.31 Prod
small change
to be Mulk mHealth
i
1400N
offs