September 22

• Still trying to draw a bicycle!
• Pack squares with side \(1/2, 1/3, 1/4,...\) into the square with side \(1\).
• Find products of some triples of consecutive numbers. Are they divisible by \(6\)? Is this always so?
• What last digit of a product depends on? Find last digits of some pairs, like \(23\times 56\) and \(53\times 36\). Are they always the same? What is the last digit of \(847281937\times 938274\times 9629052717\)?
• How the remainder of division of \(n\) by \(10\) is related to the last digit of \(n\)? What is \(n\) is negative?
• Find the last digit of \(17^{17}\).
• In general, what happens with the remainders \(\mod p\) for some \(p\), like \(5\) or \(9\), when numbers are multiplied? 
• Sets: intersections (AND) and unions (OR). 

Homework:

• If one finds the sum of digits of \(17^{17}\), then sum of its digits, and so on, till one digit remains, - what is it?
• The following sets are given:\[A = \{1, 3, 7, 137\}, B = \{3, 7, 100\}, C = \{0, 1, 3, 100\}, D = \{0, 7, 100, 333\}. \]
  Describe the sets:
• \(A \cup B\);
• \(A\cap B\);
• \((A \cap B)\cup D\);
• \(C \cap (D\cap B)\);
• \((A\cup B)\cap (C \cup D)\);
• \((A\cap B)\cup (C \cap D)\);
• \((D\cup A)\cap (C \cup B)\);
• \((A\cap (B\cap C))\cap D\);
• \((A\cup (B\cap C))\cap D\);
• \((C \cap A)\cup ((A\cup (C \cap D))\cap B)\)

Reminder: no class on Sept. 29.