MATH1061/7861, Mon 24 Aug 2020 Question 1. Prove, disprove, or salvage if possible:
For any integers a and b, if a | b then a ∤(b+1).
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◦ Determine all the integers between 10 and 20 that are congruent to −42 modulo 5.
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Question 3. Let n be any odd integer.
◦ Prove that n2 mod 4 = 1.
◦ Write a formula for⎣n2/4⎦without using the floor or ceiling symbols.
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Question 4. Suppose that a mod 6 = 4 and b mod 6 = 3. ◦ Determine (a + b) mod 6.
◦ Determine ab mod 6. 2
◦ Determine (5a2 + 2b) mod 6.
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Question 5. Let n be any positive integer.
◦ Prove that 3 | n if and only if the sum of the digits of n is divisible by 3.
◦ Prove that 9 | n if and only if the sum of the digits of n is divisible by 9.
◦ Prove that 11 | n if and only if the alternating sum of the digits of n is divisible by 11.
The alternating sum of a sequence a, b, c, d, e, ... is the sum (a − b + c − d + e − ...). Theorem. There are infinitely many primes.
Challenge (Last week). Prove the following statement:
For all m ∈ N, if m, m + 2 and m + 4 are all prime, then m = 3.
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