MATH221 Mathematics for Computer ScienceTutorial Sheet Week 7 – Autumn 20211. A penny collection contains twelve 1967 pennies, seven 1968 pennies, and eleven 1971 pennies. If you are topick some pennies without looking at the dates, how many must you pick to be sure of getting at least five penniesfrom the same year?2. (i) How many integers must you pick in order to be sure that at least two of them have the same remainderwhen divided by 15?(ii) How many integers between 100 and 999 must you pick in order to be sure that at least two of them have adigit in common?3. (i) Find values for m; n 2 Z such that gcd(98; 85) = 98m + 85n.(ii) Find values for m; n 2 Z such that gcd(224; 1715) = 224m + 1715n.4. Let a; b 2 N. Show that a and b are coprime if and only if no prime p divides both a and b (that is, a and bhave no common prime factor).5. Write down three pairs of integers, all with at least two digits, that are relatively prime. Make sure that someof the numbers you pick are not primes. Explain how you know that they are relatively prime. (Question 4 maybe useful; don’t use the Euclidean algorithm.)6. Find numbers in the range 0; 1; : : : ; 11 to which the following numbers are congruent modulo 12.-17; 3; 43; 15; 37; -11; -217. Find all solutions m 2 Z of the equations 6 ≡ m (mod 3) and 125 ≡ m (mod 7).8. Find 2100 mod 5 and 3100 mod 7 using the methods discussed in lectures.9. Let n 2 N. Prove that for all a; b 2 Z, [a] = [b] () a ≡ b (mod n).10. Write out the addition and multiplication tables for Z6. Do all elements in Z6 have multiplicative inverses?For m = 1; 2; 3; 4; 5, write down gcd(m; 6) and confirm that [m] 2 Z6 has an inverse under multiplication if andonly if gcd(m; 6) = 1.11. (i) For m 2 Z, is [m] a number, a set of numbers or a set of sets? Is Z7 a number, a set of numbers or a setof sets?(ii) Describe [2], [5] and [7] in Z7 by listing eight elements in each set. Write down the set Z7.(iii) Are the following statements true or false (or perhaps meaningless)?(a) [2] = [9] (b) [5] = 5 (mod 7) (c) 5 2 [19](d) 3 2 Z7 (e) [10] 2 Z7 (f) a ≡ b (mod 7) ) [a] 2 [b]

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