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.
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Question 2. Prove or disprove: ∀a,b,c ∈ Z, if a|(b+c) then a|b or a|c.
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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
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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
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