MATH135 Lecture Notes - Lecture 6: Fax, Prime Number, Perfect Number
14 May 2018
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MATH 135 Fall 2015: Extra Practice Set 6
These problems are for extra practice and are not to be handed. Solutions will not be posted but, unlike
assignment problems, they may discussed in depth on Piazza.
•The warm-up exercises are intended to be fairly quick and easy to solve. If you are unsure about any
of them, then you should review your notes and possibly speak to an instructor before beginning the
corresponding assignment.
•The recommended problems supplement the practice gained by doing the corresponding assignment.
Some should be done as the material is learned and the rest can be left for exam preparation.
•A few more challenging extra problems are also included for students wishing to push themselves
even harder. Do not worry if you cannot solve these more difficult problems.
Warm-up Exercises
1. Let a, b, c ∈Z. Disprove the statement: If a|(bc), then a|bor a|c.
2. Find the complete solution to 7x+ 11y= 3.
3. Find the complete solution to 28x+ 60y= 10.
4. What is the smallest non-negative integer xsuch that 2000 ≡x(mod 37)?
Recommended Problems
1. How many positive divisors does 33480 have?
2. Find all non-negative integer solutions to 12x+ 57y= 423.
3. Prove or disprove the following statements. Let a, b, c be fixed integers.
(a) If there exists an integer solution to ax2+by2=c, then gcd(a, b)|c.
(b) If gcd(a, b)|c, then there exists an integer solution to ax2+by2=c.
4. Prove or disprove: If 7a2=b2where a, b ∈Z, then 7 is a common divisor of aand b.
5. Prove that if pis prime and p≤n, then pdoes not divide n! + 1.
6. For what values of cdoes 8x+ 5y=chave exactly one solution where both xand yare strictly
positive?
7. Consider the following statement:
Let a, b, c ∈Z. For every integer x0, there exists an integer y0such that ax0+by0=c.
(a) Determine conditions on a, b, c such that the statement is true if and only if these conditions
hold. State and prove this if and only if statement.
(b) Carefully write down the negation of the given statement and prove that this negation is true.
8. Suppose aand bare integers. Prove that {ax +by |x, y ∈Z}={n·gcd(a, b)|n∈Z}.
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Document Summary
Math 135 fall 2015: extra practice set 6. These problems are for extra practice and are not to be handed. Solutions will not be posted but, unlike assignment problems, they may discussed in depth on piazza: the warm-up exercises are intended to be fairly quick and easy to solve. If you are unsure about any of them, then you should review your notes and possibly speak to an instructor before beginning the corresponding assignment: the recommended problems supplement the practice gained by doing the corresponding assignment. Some should be done as the material is learned and the rest can be left for exam preparation: a few more challenging extra problems are also included for students wishing to push themselves even harder. Do not worry if you cannot solve these more di cult problems. Recommended problems: how many positive divisors does 33480 have, find all non-negative integer solutions to 12x + 57y = 423, prove or disprove the following statements.
